[{"data":1,"prerenderedAt":24164},["ShallowReactive",2],{"locale-switcher-blog-es-null":3,"home-es":4,"home-recent-es":64},null,{"id":5,"title":6,"_locale":7,"body":8,"description":55,"extension":56,"meta":57,"navigation":58,"path":59,"seo":60,"slug":61,"stem":62,"__hash__":63},"pages\u002Fpages\u002Findex.es.md","Lanzar Moneda Online — Cara o Cruz Gratis","es",{"type":9,"value":10,"toc":50},"minimal",[11,16,20,30,41,44],[12,13,15],"h2",{"id":14},"un-lanzamiento-de-moneda-justo","Un lanzamiento de moneda justo",[17,18,19],"p",{},"El juego de la incertidumbre: lanzar una moneda nos da aproximadamente un 50% de probabilidades de que caiga en cara y un 50% de que caiga en cruz, pero hasta que la moneda aterriza, ¡no sabemos cuál será!",[17,21,22,23,29],{},"Curiosamente, los lanzamientos de moneda reales hechos por humanos no siempre son perfectamente justos. Los estudios han demostrado que una moneda tiene alrededor de un 51% de probabilidad de caer sobre la misma cara con la que empezó. Factores como la resistencia del aire, la distribución de peso de la moneda e incluso el movimiento mecánico del lanzamiento pueden afectar el resultado. Por ejemplo, un ",[24,25,28],"a",{"href":26,"target":27},"https:\u002F\u002Fwww.stat.berkeley.edu\u002F~aldous\u002F157\u002FPapers\u002Fdiaconis_coinbias.pdf","_blank"," (PDF) estudio de 2007 de los matemáticos Persi Diaconis, Susan Holmes y Richard Montgomery (DHM)"," encontró que la probabilidad de caer sobre la misma cara era del 51% para lanzamientos naturales.",[17,31,32,36,37,40],{},[33,34,35],"strong",{},"En teoría",", una moneda es justa y el resultado es aleatorio. Pero ",[33,38,39],{},"en la práctica, no lo es",".",[17,42,43],{},"Cada lanzamiento de moneda es un \"suceso\" independiente y el resultado del lanzamiento anterior no afecta a la probabilidad del siguiente. Pero, como hemos descrito, el lanzamiento real no es tan justo como nos gustaría.",[17,45,46,49],{},[33,47,48],{},"En flipthecoin.app",", intentamos que el lanzamiento de la moneda sea lo más justo posible introduciendo un componente de aleatoriedad para que no sea determinista. Nuestro lanzamiento no puede trucarse tan fácilmente.",{"title":51,"searchDepth":52,"depth":52,"links":53},"",2,[54],{"id":14,"depth":52,"text":15},"Lanza una moneda online con nuestro simulador 3D gratuito. Resultado instantáneo cara o cruz — sin instalar nada. Justo, aleatorio y con física real.","md",{},true,"\u002Fpages\u002Findex.es",{"title":6,"description":55},"index","pages\u002Findex.es","XbgKX3gCXVHKEbHNBXaoCw6DSXYo0RDAaD2PA8FOx34",[65,13174,23727],{"id":66,"title":67,"_locale":7,"body":68,"date":13020,"description":13021,"extension":56,"meta":13022,"navigation":58,"path":13169,"published":58,"seo":13170,"slug":13171,"stem":13172,"__hash__":13173},"blog\u002Fblog\u002Fpi-coin-flips-catalan-numbers.es.md","Lanzando una moneda hasta que obtenemos PI",{"type":9,"value":69,"toc":13005},[70,74,232,271,280,283,304,307,614,618,634,639,642,683,903,907,914,959,1317,1321,1328,1335,1338,1347,1427,1553,1556,1562,1681,1718,1856,2204,2208,2215,2219,3597,3600,4023,4121,4124,4128,4266,4295,4298,4330,4603,4913,4916,5425,5932,6196,6203,6207,6409,6413,6472,7275,7334,7775,7778,9213,9386,9390,9393,9653,9918,10439,10442,11215,11222,11724,11813,11868,12236,12348,12488,12491,12799,12844,12847,12883,12889,12893,12896,12927,12959,12962,12994,12997],[71,72,67],"h1",{"id":73},"lanzando-una-moneda-hasta-que-obtenemos-pi",[17,75,76,77,83,84,89,90,40],{},"Matt Parker, del canal ",[24,78,82],{"href":79,"rel":80},"https:\u002F\u002Fwww.youtube.com\u002F@standupmaths",[81],"nofollow","Stand-up Maths",", publicó recientemente ",[24,85,88],{"href":86,"rel":87},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=kahGSss6SsU",[81],"un vídeo"," con una premisa tan simple de enunciar como difícil de creer: si lanzamos una moneda repetidamente hasta que haya más caras que cruces, y calculamos la proporción media de caras respecto al total de lanzamientos... el resultado es ",[91,92,95,125],"span",{"className":93},[94],"katex",[91,96,99],{"className":97},[98],"katex-mathml",[100,101,103],"math",{"xmlns":102},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[104,105,106,120],"semantics",{},[107,108,109],"mrow",{},[110,111,112,116],"mfrac",{},[113,114,115],"mi",{},"π",[117,118,119],"mn",{},"4",[121,122,124],"annotation",{"encoding":123},"application\u002Fx-tex","\\frac{\\pi}{4}",[91,126,130],{"className":127,"ariaHidden":129},[128],"katex-html","true",[91,131,134,139],{"className":132},[133],"base",[91,135],{"className":136,"style":138},[137],"strut","height:1.0404em;vertical-align:-0.345em;",[91,140,143,148,228],{"className":141},[142],"mord",[91,144],{"className":145},[146,147],"mopen","nulldelimiter",[91,149,151],{"className":150},[110],[91,152,156,219],{"className":153},[154,155],"vlist-t","vlist-t2",[91,157,160,214],{"className":158},[159],"vlist-r",[91,161,165,186,197],{"className":162,"style":164},[163],"vlist","height:0.6954em;",[91,166,168,173],{"style":167},"top:-2.655em;",[91,169],{"className":170,"style":172},[171],"pstrut","height:3em;",[91,174,180],{"className":175},[176,177,178,179],"sizing","reset-size6","size3","mtight",[91,181,183],{"className":182},[142,179],[91,184,119],{"className":185},[142,179],[91,187,189,192],{"style":188},"top:-3.23em;",[91,190],{"className":191,"style":172},[171],[91,193],{"className":194,"style":196},[195],"frac-line","border-bottom-width:0.04em;",[91,198,200,203],{"style":199},"top:-3.394em;",[91,201],{"className":202,"style":172},[171],[91,204,206],{"className":205},[176,177,178,179],[91,207,209],{"className":208},[142,179],[91,210,115],{"className":211,"style":213},[142,212,179],"mathnormal","margin-right:0.0359em;",[91,215,218],{"className":216},[217],"vlist-s","​",[91,220,222],{"className":221},[159],[91,223,226],{"className":224,"style":225},[163],"height:0.345em;",[91,227],{},[91,229],{"className":230},[231,147],"mclose",[17,233,234,235,265,266,270],{},"Sí, ",[91,236,238,252],{"className":237},[94],[91,239,241],{"className":240},[98],[100,242,243],{"xmlns":102},[104,244,245,249],{},[107,246,247],{},[113,248,115],{},[121,250,251],{"encoding":123},"\\pi",[91,253,255],{"className":254,"ariaHidden":129},[128],[91,256,258,262],{"className":257},[133],[91,259],{"className":260,"style":261},[137],"height:0.4306em;",[91,263,115],{"className":264,"style":213},[142,212],". ",[267,268,269],"em",{},"El número de los círculos",". Escondido dentro de lanzamientos de moneda.",[17,272,273,274,279],{},"En nuestro ",[24,275,278],{"href":276,"rel":277},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Ftriangulo-de-tartaglia-combinatoria-y-probabilidad\u002F",[81],"artículo anterior sobre el triángulo de Tartaglia"," vimos cómo un triángulo de sumas sencillas contenía, en su interior, la combinatoria, el álgebra y la probabilidad. Hoy vamos a tirar del mismo hilo. Y nos va a llevar a un lugar que nadie esperaba.",[17,281,282],{},"Las reglas del experimento son sencillas:",[284,285,286,290,297],"ol",{},[287,288,289],"li",{},"Lanzamos una moneda justa repetidamente.",[287,291,292,293,296],{},"Nos detenemos ",[33,294,295],{},"en el momento exacto"," en que el número de caras supera al de cruces.",[287,298,299,300,303],{},"Anotamos la ",[33,301,302],{},"proporción de caras"," respecto al total de lanzamientos de esa secuencia.",[17,305,306],{},"La pregunta del artículo será: si repetimos este proceso infinitas veces y calculamos la media de todas esas proporciones... ¿qué obtenemos?",[17,308,309,310,439,440,539,540,584,585,613],{},"Antes de responder, pensemos en qué rango puede moverse esa proporción. En el mejor de los casos, el primer lanzamiento es cara: paramos inmediatamente y la proporción es ",[91,311,313,338],{"className":312},[94],[91,314,316],{"className":315},[98],[100,317,318],{"xmlns":102},[104,319,320,335],{},[107,321,322,329,333],{},[110,323,324,327],{},[117,325,326],{},"1",[117,328,326],{},[330,331,332],"mo",{},"=",[117,334,326],{},[121,336,337],{"encoding":123},"\\frac{1}{1} = 1",[91,339,341,429],{"className":340,"ariaHidden":129},[128],[91,342,344,348,417,422,426],{"className":343},[133],[91,345],{"className":346,"style":347},[137],"height:1.1901em;vertical-align:-0.345em;",[91,349,351,354,414],{"className":350},[142],[91,352],{"className":353},[146,147],[91,355,357],{"className":356},[110],[91,358,360,406],{"className":359},[154,155],[91,361,363,403],{"className":362},[159],[91,364,367,381,389],{"className":365,"style":366},[163],"height:0.8451em;",[91,368,369,372],{"style":167},[91,370],{"className":371,"style":172},[171],[91,373,375],{"className":374},[176,177,178,179],[91,376,378],{"className":377},[142,179],[91,379,326],{"className":380},[142,179],[91,382,383,386],{"style":188},[91,384],{"className":385,"style":172},[171],[91,387],{"className":388,"style":196},[195],[91,390,391,394],{"style":199},[91,392],{"className":393,"style":172},[171],[91,395,397],{"className":396},[176,177,178,179],[91,398,400],{"className":399},[142,179],[91,401,326],{"className":402},[142,179],[91,404,218],{"className":405},[217],[91,407,409],{"className":408},[159],[91,410,412],{"className":411,"style":225},[163],[91,413],{},[91,415],{"className":416},[231,147],[91,418],{"className":419,"style":421},[420],"mspace","margin-right:0.2778em;",[91,423,332],{"className":424},[425],"mrel",[91,427],{"className":428,"style":421},[420],[91,430,432,436],{"className":431},[133],[91,433],{"className":434,"style":435},[137],"height:0.6444em;",[91,437,326],{"className":438},[142]," (el 100 % son caras). En el peor, necesitamos miles de lanzamientos hasta que las caras superen a las cruces por uno, y la proporción será apenas superior a ",[91,441,443,462],{"className":442},[94],[91,444,446],{"className":445},[98],[100,447,448],{"xmlns":102},[104,449,450,459],{},[107,451,452],{},[110,453,454,456],{},[117,455,326],{},[117,457,458],{},"2",[121,460,461],{"encoding":123},"\\frac{1}{2}",[91,463,465],{"className":464,"ariaHidden":129},[128],[91,466,468,471],{"className":467},[133],[91,469],{"className":470,"style":347},[137],[91,472,474,477,536],{"className":473},[142],[91,475],{"className":476},[146,147],[91,478,480],{"className":479},[110],[91,481,483,528],{"className":482},[154,155],[91,484,486,525],{"className":485},[159],[91,487,489,503,511],{"className":488,"style":366},[163],[91,490,491,494],{"style":167},[91,492],{"className":493,"style":172},[171],[91,495,497],{"className":496},[176,177,178,179],[91,498,500],{"className":499},[142,179],[91,501,458],{"className":502},[142,179],[91,504,505,508],{"style":188},[91,506],{"className":507,"style":172},[171],[91,509],{"className":510,"style":196},[195],[91,512,513,516],{"style":199},[91,514],{"className":515,"style":172},[171],[91,517,519],{"className":518},[176,177,178,179],[91,520,522],{"className":521},[142,179],[91,523,326],{"className":524},[142,179],[91,526,218],{"className":527},[217],[91,529,531],{"className":530},[159],[91,532,534],{"className":533,"style":225},[163],[91,535],{},[91,537],{"className":538},[231,147],". Así que la media estará en algún punto entre ",[91,541,543,558],{"className":542},[94],[91,544,546],{"className":545},[98],[100,547,548],{"xmlns":102},[104,549,550,555],{},[107,551,552],{},[117,553,554],{},"0,5",[121,556,557],{"encoding":123},"0{,}5",[91,559,561],{"className":560,"ariaHidden":129},[128],[91,562,564,568,572,580],{"className":563},[133],[91,565],{"className":566,"style":567},[137],"height:0.8389em;vertical-align:-0.1944em;",[91,569,571],{"className":570},[142],"0",[91,573,575],{"className":574},[142],[91,576,579],{"className":577},[578],"mpunct",",",[91,581,583],{"className":582},[142],"5"," y ",[91,586,588,601],{"className":587},[94],[91,589,591],{"className":590},[98],[100,592,593],{"xmlns":102},[104,594,595,599],{},[107,596,597],{},[117,598,326],{},[121,600,326],{"encoding":123},[91,602,604],{"className":603,"ariaHidden":129},[128],[91,605,607,610],{"className":606},[133],[91,608],{"className":609,"style":435},[137],[91,611,326],{"className":612},[142],". Veamos exactamente dónde.",[12,615,617],{"id":616},"construyendo-las-secuencias-paso-a-paso","Construyendo las secuencias, paso a paso",[17,619,620,621,624,625,628,629,40],{},"Vamos a construir las secuencias posibles empezando por las más cortas. Usaremos ",[33,622,623],{},"O"," para cara y ",[33,626,627],{},"X"," para cruz — la misma notación que usamos en ",[24,630,633],{"href":631,"rel":632},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fplay\u002F",[81],"nuestra herramienta de lanzamiento",[635,636,638],"h3",{"id":637},"longitud-1","Longitud 1",[17,640,641],{},"La secuencia más corta posible: lanzamos una vez y sale cara.",[17,643,646],{"className":644},[645],"math-block",[91,647,649,665],{"className":648},[94],[91,650,652],{"className":651},[98],[100,653,654],{"xmlns":102},[104,655,656,662],{},[107,657,658],{},[659,660,623],"mtext",{"mathvariant":661},"bold",[121,663,664],{"encoding":123},"\\textbf{O}",[91,666,668],{"className":667,"ariaHidden":129},[128],[91,669,671,675],{"className":670},[133],[91,672],{"className":673,"style":674},[137],"height:0.6861em;",[91,676,679],{"className":677},[142,678],"text",[91,680,623],{"className":681},[142,682],"textbf",[17,684,685,686,805,806,40],{},"Paramos. Tenemos 1 cara y 0 cruces. La proporción es ",[91,687,689,710],{"className":688},[94],[91,690,692],{"className":691},[98],[100,693,694],{"xmlns":102},[104,695,696,708],{},[107,697,698,704,706],{},[110,699,700,702],{},[117,701,326],{},[117,703,326],{},[330,705,332],{},[117,707,326],{},[121,709,337],{"encoding":123},[91,711,713,796],{"className":712,"ariaHidden":129},[128],[91,714,716,719,787,790,793],{"className":715},[133],[91,717],{"className":718,"style":347},[137],[91,720,722,725,784],{"className":721},[142],[91,723],{"className":724},[146,147],[91,726,728],{"className":727},[110],[91,729,731,776],{"className":730},[154,155],[91,732,734,773],{"className":733},[159],[91,735,737,751,759],{"className":736,"style":366},[163],[91,738,739,742],{"style":167},[91,740],{"className":741,"style":172},[171],[91,743,745],{"className":744},[176,177,178,179],[91,746,748],{"className":747},[142,179],[91,749,326],{"className":750},[142,179],[91,752,753,756],{"style":188},[91,754],{"className":755,"style":172},[171],[91,757],{"className":758,"style":196},[195],[91,760,761,764],{"style":199},[91,762],{"className":763,"style":172},[171],[91,765,767],{"className":766},[176,177,178,179],[91,768,770],{"className":769},[142,179],[91,771,326],{"className":772},[142,179],[91,774,218],{"className":775},[217],[91,777,779],{"className":778},[159],[91,780,782],{"className":781,"style":225},[163],[91,783],{},[91,785],{"className":786},[231,147],[91,788],{"className":789,"style":421},[420],[91,791,332],{"className":792},[425],[91,794],{"className":795,"style":421},[420],[91,797,799,802],{"className":798},[133],[91,800],{"className":801,"style":435},[137],[91,803,326],{"className":804},[142]," y la probabilidad de que esto ocurra es ",[91,807,809,826],{"className":808},[94],[91,810,812],{"className":811},[98],[100,813,814],{"xmlns":102},[104,815,816,824],{},[107,817,818],{},[110,819,820,822],{},[117,821,326],{},[117,823,458],{},[121,825,461],{"encoding":123},[91,827,829],{"className":828,"ariaHidden":129},[128],[91,830,832,835],{"className":831},[133],[91,833],{"className":834,"style":347},[137],[91,836,838,841,900],{"className":837},[142],[91,839],{"className":840},[146,147],[91,842,844],{"className":843},[110],[91,845,847,892],{"className":846},[154,155],[91,848,850,889],{"className":849},[159],[91,851,853,867,875],{"className":852,"style":366},[163],[91,854,855,858],{"style":167},[91,856],{"className":857,"style":172},[171],[91,859,861],{"className":860},[176,177,178,179],[91,862,864],{"className":863},[142,179],[91,865,458],{"className":866},[142,179],[91,868,869,872],{"style":188},[91,870],{"className":871,"style":172},[171],[91,873],{"className":874,"style":196},[195],[91,876,877,880],{"style":199},[91,878],{"className":879,"style":172},[171],[91,881,883],{"className":882},[176,177,178,179],[91,884,886],{"className":885},[142,179],[91,887,326],{"className":888},[142,179],[91,890,218],{"className":891},[217],[91,893,895],{"className":894},[159],[91,896,898],{"className":897,"style":225},[163],[91,899],{},[91,901],{"className":902},[231,147],[635,904,906],{"id":905},"longitud-3","Longitud 3",[17,908,909,910,913],{},"Si el primer lanzamiento es cruz, no podemos parar. Y con sólo dos lanzamientos, lo mejor que podemos lograr es empatar (1 cara, 1 cruz) — empatar no basta, necesitamos que las caras ",[267,911,912],{},"superen"," a las cruces. Así que la siguiente secuencia posible tiene longitud 3:",[17,915,917],{"className":916},[645],[91,918,920,943],{"className":919},[94],[91,921,923],{"className":922},[98],[100,924,925],{"xmlns":102},[104,926,927,940],{},[107,928,929,931,934,936,938],{},[659,930,627],{"mathvariant":661},[659,932,933],{}," ",[659,935,623],{"mathvariant":661},[659,937,933],{},[659,939,623],{"mathvariant":661},[121,941,942],{"encoding":123},"\\textbf{X O O}",[91,944,946],{"className":945,"ariaHidden":129},[128],[91,947,949,952],{"className":948},[133],[91,950],{"className":951,"style":674},[137],[91,953,955],{"className":954},[142,678],[91,956,958],{"className":957},[142,682],"X O O",[17,960,961,962,1061,1062,1316],{},"Cruz, cara, cara. El tercer lanzamiento nos da 2 caras frente a 1 cruz. Proporción: ",[91,963,965,984],{"className":964},[94],[91,966,968],{"className":967},[98],[100,969,970],{"xmlns":102},[104,971,972,981],{},[107,973,974],{},[110,975,976,978],{},[117,977,458],{},[117,979,980],{},"3",[121,982,983],{"encoding":123},"\\frac{2}{3}",[91,985,987],{"className":986,"ariaHidden":129},[128],[91,988,990,993],{"className":989},[133],[91,991],{"className":992,"style":347},[137],[91,994,996,999,1058],{"className":995},[142],[91,997],{"className":998},[146,147],[91,1000,1002],{"className":1001},[110],[91,1003,1005,1050],{"className":1004},[154,155],[91,1006,1008,1047],{"className":1007},[159],[91,1009,1011,1025,1033],{"className":1010,"style":366},[163],[91,1012,1013,1016],{"style":167},[91,1014],{"className":1015,"style":172},[171],[91,1017,1019],{"className":1018},[176,177,178,179],[91,1020,1022],{"className":1021},[142,179],[91,1023,980],{"className":1024},[142,179],[91,1026,1027,1030],{"style":188},[91,1028],{"className":1029,"style":172},[171],[91,1031],{"className":1032,"style":196},[195],[91,1034,1035,1038],{"style":199},[91,1036],{"className":1037,"style":172},[171],[91,1039,1041],{"className":1040},[176,177,178,179],[91,1042,1044],{"className":1043},[142,179],[91,1045,458],{"className":1046},[142,179],[91,1048,218],{"className":1049},[217],[91,1051,1053],{"className":1052},[159],[91,1054,1056],{"className":1055,"style":225},[163],[91,1057],{},[91,1059],{"className":1060},[231,147]," (2 caras de 3 lanzamientos). Probabilidad: ",[91,1063,1065,1105],{"className":1064},[94],[91,1066,1068],{"className":1067},[98],[100,1069,1070],{"xmlns":102},[104,1071,1072,1102],{},[107,1073,1074,1093,1095],{},[1075,1076,1077,1091],"msup",{},[107,1078,1079,1082,1088],{},[330,1080,1081],{"fence":129},"(",[110,1083,1084,1086],{},[117,1085,326],{},[117,1087,458],{},[330,1089,1090],{"fence":129},")",[117,1092,980],{},[330,1094,332],{},[110,1096,1097,1099],{},[117,1098,326],{},[117,1100,1101],{},"8",[121,1103,1104],{"encoding":123},"\\left(\\frac{1}{2}\\right)^3 = \\frac{1}{8}",[91,1106,1108,1242],{"className":1107,"ariaHidden":129},[128],[91,1109,1111,1115,1233,1236,1239],{"className":1110},[133],[91,1112],{"className":1113,"style":1114},[137],"height:1.404em;vertical-align:-0.35em;",[91,1116,1119,1206],{"className":1117},[1118],"minner",[91,1120,1122,1132,1200],{"className":1121},[1118],[91,1123,1127],{"className":1124,"style":1126},[146,1125],"delimcenter","top:0em;",[91,1128,1081],{"className":1129},[1130,1131],"delimsizing","size1",[91,1133,1135,1138,1197],{"className":1134},[142],[91,1136],{"className":1137},[146,147],[91,1139,1141],{"className":1140},[110],[91,1142,1144,1189],{"className":1143},[154,155],[91,1145,1147,1186],{"className":1146},[159],[91,1148,1150,1164,1172],{"className":1149,"style":366},[163],[91,1151,1152,1155],{"style":167},[91,1153],{"className":1154,"style":172},[171],[91,1156,1158],{"className":1157},[176,177,178,179],[91,1159,1161],{"className":1160},[142,179],[91,1162,458],{"className":1163},[142,179],[91,1165,1166,1169],{"style":188},[91,1167],{"className":1168,"style":172},[171],[91,1170],{"className":1171,"style":196},[195],[91,1173,1174,1177],{"style":199},[91,1175],{"className":1176,"style":172},[171],[91,1178,1180],{"className":1179},[176,177,178,179],[91,1181,1183],{"className":1182},[142,179],[91,1184,326],{"className":1185},[142,179],[91,1187,218],{"className":1188},[217],[91,1190,1192],{"className":1191},[159],[91,1193,1195],{"className":1194,"style":225},[163],[91,1196],{},[91,1198],{"className":1199},[231,147],[91,1201,1203],{"className":1202,"style":1126},[231,1125],[91,1204,1090],{"className":1205},[1130,1131],[91,1207,1210],{"className":1208},[1209],"msupsub",[91,1211,1213],{"className":1212},[154],[91,1214,1216],{"className":1215},[159],[91,1217,1220],{"className":1218,"style":1219},[163],"height:1.054em;",[91,1221,1223,1227],{"style":1222},"top:-3.3029em;margin-right:0.05em;",[91,1224],{"className":1225,"style":1226},[171],"height:2.7em;",[91,1228,1230],{"className":1229},[176,177,178,179],[91,1231,980],{"className":1232},[142,179],[91,1234],{"className":1235,"style":421},[420],[91,1237,332],{"className":1238},[425],[91,1240],{"className":1241,"style":421},[420],[91,1243,1245,1248],{"className":1244},[133],[91,1246],{"className":1247,"style":347},[137],[91,1249,1251,1254,1313],{"className":1250},[142],[91,1252],{"className":1253},[146,147],[91,1255,1257],{"className":1256},[110],[91,1258,1260,1305],{"className":1259},[154,155],[91,1261,1263,1302],{"className":1262},[159],[91,1264,1266,1280,1288],{"className":1265,"style":366},[163],[91,1267,1268,1271],{"style":167},[91,1269],{"className":1270,"style":172},[171],[91,1272,1274],{"className":1273},[176,177,178,179],[91,1275,1277],{"className":1276},[142,179],[91,1278,1101],{"className":1279},[142,179],[91,1281,1282,1285],{"style":188},[91,1283],{"className":1284,"style":172},[171],[91,1286],{"className":1287,"style":196},[195],[91,1289,1290,1293],{"style":199},[91,1291],{"className":1292,"style":172},[171],[91,1294,1296],{"className":1295},[176,177,178,179],[91,1297,1299],{"className":1298},[142,179],[91,1300,326],{"className":1301},[142,179],[91,1303,218],{"className":1304},[217],[91,1306,1308],{"className":1307},[159],[91,1309,1311],{"className":1310,"style":225},[163],[91,1312],{},[91,1314],{"className":1315},[231,147],". Y es la única secuencia posible de esta longitud.",[635,1318,1320],{"id":1319},"longitud-5","Longitud 5",[17,1322,1323,1324,1327],{},"Aquí se pone interesante. Necesitamos 3 caras y 2 cruces, y el quinto lanzamiento (el que nos detiene) tiene que ser cara. Además — y esto es clave — ",[33,1325,1326],{},"en ningún momento antes del final podemos haber tenido más caras que cruces",", porque si eso hubiera ocurrido, nos habríamos detenido antes.",[17,1329,1330,1331,1334],{},"No basta con que la secuencia termine con más caras que cruces; tiene que ser ",[33,1332,1333],{},"la primera vez"," que ocurre.",[17,1336,1337],{},"¿Cuántas secuencias cumplen estas condiciones? Sólo dos:",[1339,1340,1344],"pre",{"className":1341,"code":1343,"language":678,"meta":51},[1342],"language-text","X X O O O\nX O X O O\n",[1345,1346,1343],"code",{"__ignoreMap":51},[17,1348,1349,1350,1353,1354,1390,1391,1426],{},"Comprobemos la primera: ",[33,1351,1352],{},"XXOOO",". Si contamos O como ",[91,1355,1357,1374],{"className":1356},[94],[91,1358,1360],{"className":1359},[98],[100,1361,1362],{"xmlns":102},[104,1363,1364,1371],{},[107,1365,1366,1369],{},[330,1367,1368],{},"+",[117,1370,326],{},[121,1372,1373],{"encoding":123},"+1",[91,1375,1377],{"className":1376,"ariaHidden":129},[128],[91,1378,1380,1384,1387],{"className":1379},[133],[91,1381],{"className":1382,"style":1383},[137],"height:0.7278em;vertical-align:-0.0833em;",[91,1385,1368],{"className":1386},[142],[91,1388,326],{"className":1389},[142]," y X como ",[91,1392,1394,1411],{"className":1393},[94],[91,1395,1397],{"className":1396},[98],[100,1398,1399],{"xmlns":102},[104,1400,1401,1408],{},[107,1402,1403,1406],{},[330,1404,1405],{},"−",[117,1407,326],{},[121,1409,1410],{"encoding":123},"-1",[91,1412,1414],{"className":1413,"ariaHidden":129},[128],[91,1415,1417,1420,1423],{"className":1416},[133],[91,1418],{"className":1419,"style":1383},[137],[91,1421,1405],{"className":1422},[142],[91,1424,326],{"className":1425},[142],", las sumas parciales son:",[17,1428,1430],{"className":1429},[645],[91,1431,1433,1480],{"className":1432},[94],[91,1434,1436],{"className":1435},[98],[100,1437,1438],{"xmlns":102},[104,1439,1440,1477],{},[107,1441,1442,1444,1446,1448,1451,1453,1455,1457,1459,1461,1463,1465,1467,1469,1471,1473,1475],{},[330,1443,1405],{},[117,1445,326],{},[330,1447,579],{"separator":129},[659,1449,1450],{},"  ",[330,1452,1405],{},[117,1454,458],{},[330,1456,579],{"separator":129},[659,1458,1450],{},[330,1460,1405],{},[117,1462,326],{},[330,1464,579],{"separator":129},[659,1466,1450],{},[117,1468,571],{},[330,1470,579],{"separator":129},[659,1472,1450],{},[330,1474,1368],{},[117,1476,326],{},[121,1478,1479],{"encoding":123},"-1,\\; -2,\\; -1,\\; 0,\\; +1",[91,1481,1483],{"className":1482,"ariaHidden":129},[128],[91,1484,1486,1489,1492,1495,1498,1501,1505,1508,1511,1514,1517,1520,1523,1526,1529,1532,1535,1538,1541,1544,1547,1550],{"className":1485},[133],[91,1487],{"className":1488,"style":567},[137],[91,1490,1405],{"className":1491},[142],[91,1493,326],{"className":1494},[142],[91,1496,579],{"className":1497},[578],[91,1499],{"className":1500,"style":421},[420],[91,1502],{"className":1503,"style":1504},[420],"margin-right:0.1667em;",[91,1506,1405],{"className":1507},[142],[91,1509,458],{"className":1510},[142],[91,1512,579],{"className":1513},[578],[91,1515],{"className":1516,"style":421},[420],[91,1518],{"className":1519,"style":1504},[420],[91,1521,1405],{"className":1522},[142],[91,1524,326],{"className":1525},[142],[91,1527,579],{"className":1528},[578],[91,1530],{"className":1531,"style":421},[420],[91,1533],{"className":1534,"style":1504},[420],[91,1536,571],{"className":1537},[142],[91,1539,579],{"className":1540},[578],[91,1542],{"className":1543,"style":421},[420],[91,1545],{"className":1546,"style":1504},[420],[91,1548,1368],{"className":1549},[142],[91,1551,326],{"className":1552},[142],[17,1554,1555],{},"Nunca supera el cero hasta el quinto lanzamiento. ✅",[17,1557,1558,1559,40],{},"Ahora la segunda: ",[33,1560,1561],{},"XOXOO",[17,1563,1565],{"className":1564},[645],[91,1566,1568,1612],{"className":1567},[94],[91,1569,1571],{"className":1570},[98],[100,1572,1573],{"xmlns":102},[104,1574,1575,1609],{},[107,1576,1577,1579,1581,1583,1585,1587,1589,1591,1593,1595,1597,1599,1601,1603,1605,1607],{},[330,1578,1405],{},[117,1580,326],{},[330,1582,579],{"separator":129},[659,1584,1450],{},[117,1586,571],{},[330,1588,579],{"separator":129},[659,1590,1450],{},[330,1592,1405],{},[117,1594,326],{},[330,1596,579],{"separator":129},[659,1598,1450],{},[117,1600,571],{},[330,1602,579],{"separator":129},[659,1604,1450],{},[330,1606,1368],{},[117,1608,326],{},[121,1610,1611],{"encoding":123},"-1,\\; 0,\\; -1,\\; 0,\\; +1",[91,1613,1615],{"className":1614,"ariaHidden":129},[128],[91,1616,1618,1621,1624,1627,1630,1633,1636,1639,1642,1645,1648,1651,1654,1657,1660,1663,1666,1669,1672,1675,1678],{"className":1617},[133],[91,1619],{"className":1620,"style":567},[137],[91,1622,1405],{"className":1623},[142],[91,1625,326],{"className":1626},[142],[91,1628,579],{"className":1629},[578],[91,1631],{"className":1632,"style":421},[420],[91,1634],{"className":1635,"style":1504},[420],[91,1637,571],{"className":1638},[142],[91,1640,579],{"className":1641},[578],[91,1643],{"className":1644,"style":421},[420],[91,1646],{"className":1647,"style":1504},[420],[91,1649,1405],{"className":1650},[142],[91,1652,326],{"className":1653},[142],[91,1655,579],{"className":1656},[578],[91,1658],{"className":1659,"style":421},[420],[91,1661],{"className":1662,"style":1504},[420],[91,1664,571],{"className":1665},[142],[91,1667,579],{"className":1668},[578],[91,1670],{"className":1671,"style":421},[420],[91,1673],{"className":1674,"style":1504},[420],[91,1676,1368],{"className":1677},[142],[91,1679,326],{"className":1680},[142],[17,1682,1683,1684,1717],{},"Vuelve a cero en el segundo y cuarto lanzamiento, pero nunca lo supera. Es la primera vez que llega a ",[91,1685,1687,1702],{"className":1686},[94],[91,1688,1690],{"className":1689},[98],[100,1691,1692],{"xmlns":102},[104,1693,1694,1700],{},[107,1695,1696,1698],{},[330,1697,1368],{},[117,1699,326],{},[121,1701,1373],{"encoding":123},[91,1703,1705],{"className":1704,"ariaHidden":129},[128],[91,1706,1708,1711,1714],{"className":1707},[133],[91,1709],{"className":1710,"style":1383},[137],[91,1712,1368],{"className":1713},[142],[91,1715,326],{"className":1716},[142],". ✅",[17,1719,1720,1721,1724,1725,1821,1822,1855],{},"¿Y qué pasa con ",[33,1722,1723],{},"OXXOO","? Veamos: ",[91,1726,1728,1764],{"className":1727},[94],[91,1729,1731],{"className":1730},[98],[100,1732,1733],{"xmlns":102},[104,1734,1735,1761],{},[107,1736,1737,1739,1741,1743,1745,1747,1749,1751,1753,1755,1757,1759],{},[330,1738,1368],{},[117,1740,326],{},[330,1742,579],{"separator":129},[117,1744,571],{},[330,1746,579],{"separator":129},[330,1748,1405],{},[117,1750,326],{},[330,1752,579],{"separator":129},[117,1754,571],{},[330,1756,579],{"separator":129},[330,1758,1368],{},[117,1760,326],{},[121,1762,1763],{"encoding":123},"+1, 0, -1, 0, +1",[91,1765,1767],{"className":1766,"ariaHidden":129},[128],[91,1768,1770,1773,1776,1779,1782,1785,1788,1791,1794,1797,1800,1803,1806,1809,1812,1815,1818],{"className":1769},[133],[91,1771],{"className":1772,"style":567},[137],[91,1774,1368],{"className":1775},[142],[91,1777,326],{"className":1778},[142],[91,1780,579],{"className":1781},[578],[91,1783],{"className":1784,"style":1504},[420],[91,1786,571],{"className":1787},[142],[91,1789,579],{"className":1790},[578],[91,1792],{"className":1793,"style":1504},[420],[91,1795,1405],{"className":1796},[142],[91,1798,326],{"className":1799},[142],[91,1801,579],{"className":1802},[578],[91,1804],{"className":1805,"style":1504},[420],[91,1807,571],{"className":1808},[142],[91,1810,579],{"className":1811},[578],[91,1813],{"className":1814,"style":1504},[420],[91,1816,1368],{"className":1817},[142],[91,1819,326],{"className":1820},[142],". El primer lanzamiento ya nos pone en ",[91,1823,1825,1840],{"className":1824},[94],[91,1826,1828],{"className":1827},[98],[100,1829,1830],{"xmlns":102},[104,1831,1832,1838],{},[107,1833,1834,1836],{},[330,1835,1368],{},[117,1837,326],{},[121,1839,1373],{"encoding":123},[91,1841,1843],{"className":1842,"ariaHidden":129},[128],[91,1844,1846,1849,1852],{"className":1845},[133],[91,1847],{"className":1848,"style":1383},[137],[91,1850,1368],{"className":1851},[142],[91,1853,326],{"className":1854},[142],": nos habríamos detenido ahí. No es válida. ❌",[17,1857,1858,1859,1862,1863,2104,2105,2203],{},"Así que hay exactamente ",[33,1860,1861],{},"2 secuencias"," de longitud 5, cada una con probabilidad ",[91,1864,1866,1903],{"className":1865},[94],[91,1867,1869],{"className":1868},[98],[100,1870,1871],{"xmlns":102},[104,1872,1873,1900],{},[107,1874,1875,1891,1893],{},[1075,1876,1877,1889],{},[107,1878,1879,1881,1887],{},[330,1880,1081],{"fence":129},[110,1882,1883,1885],{},[117,1884,326],{},[117,1886,458],{},[330,1888,1090],{"fence":129},[117,1890,583],{},[330,1892,332],{},[110,1894,1895,1897],{},[117,1896,326],{},[117,1898,1899],{},"32",[121,1901,1902],{"encoding":123},"\\left(\\frac{1}{2}\\right)^5 = \\frac{1}{32}",[91,1904,1906,2030],{"className":1905,"ariaHidden":129},[128],[91,1907,1909,1912,2021,2024,2027],{"className":1908},[133],[91,1910],{"className":1911,"style":1114},[137],[91,1913,1915,1998],{"className":1914},[1118],[91,1916,1918,1924,1992],{"className":1917},[1118],[91,1919,1921],{"className":1920,"style":1126},[146,1125],[91,1922,1081],{"className":1923},[1130,1131],[91,1925,1927,1930,1989],{"className":1926},[142],[91,1928],{"className":1929},[146,147],[91,1931,1933],{"className":1932},[110],[91,1934,1936,1981],{"className":1935},[154,155],[91,1937,1939,1978],{"className":1938},[159],[91,1940,1942,1956,1964],{"className":1941,"style":366},[163],[91,1943,1944,1947],{"style":167},[91,1945],{"className":1946,"style":172},[171],[91,1948,1950],{"className":1949},[176,177,178,179],[91,1951,1953],{"className":1952},[142,179],[91,1954,458],{"className":1955},[142,179],[91,1957,1958,1961],{"style":188},[91,1959],{"className":1960,"style":172},[171],[91,1962],{"className":1963,"style":196},[195],[91,1965,1966,1969],{"style":199},[91,1967],{"className":1968,"style":172},[171],[91,1970,1972],{"className":1971},[176,177,178,179],[91,1973,1975],{"className":1974},[142,179],[91,1976,326],{"className":1977},[142,179],[91,1979,218],{"className":1980},[217],[91,1982,1984],{"className":1983},[159],[91,1985,1987],{"className":1986,"style":225},[163],[91,1988],{},[91,1990],{"className":1991},[231,147],[91,1993,1995],{"className":1994,"style":1126},[231,1125],[91,1996,1090],{"className":1997},[1130,1131],[91,1999,2001],{"className":2000},[1209],[91,2002,2004],{"className":2003},[154],[91,2005,2007],{"className":2006},[159],[91,2008,2010],{"className":2009,"style":1219},[163],[91,2011,2012,2015],{"style":1222},[91,2013],{"className":2014,"style":1226},[171],[91,2016,2018],{"className":2017},[176,177,178,179],[91,2019,583],{"className":2020},[142,179],[91,2022],{"className":2023,"style":421},[420],[91,2025,332],{"className":2026},[425],[91,2028],{"className":2029,"style":421},[420],[91,2031,2033,2036],{"className":2032},[133],[91,2034],{"className":2035,"style":347},[137],[91,2037,2039,2042,2101],{"className":2038},[142],[91,2040],{"className":2041},[146,147],[91,2043,2045],{"className":2044},[110],[91,2046,2048,2093],{"className":2047},[154,155],[91,2049,2051,2090],{"className":2050},[159],[91,2052,2054,2068,2076],{"className":2053,"style":366},[163],[91,2055,2056,2059],{"style":167},[91,2057],{"className":2058,"style":172},[171],[91,2060,2062],{"className":2061},[176,177,178,179],[91,2063,2065],{"className":2064},[142,179],[91,2066,1899],{"className":2067},[142,179],[91,2069,2070,2073],{"style":188},[91,2071],{"className":2072,"style":172},[171],[91,2074],{"className":2075,"style":196},[195],[91,2077,2078,2081],{"style":199},[91,2079],{"className":2080,"style":172},[171],[91,2082,2084],{"className":2083},[176,177,178,179],[91,2085,2087],{"className":2086},[142,179],[91,2088,326],{"className":2089},[142,179],[91,2091,218],{"className":2092},[217],[91,2094,2096],{"className":2095},[159],[91,2097,2099],{"className":2098,"style":225},[163],[91,2100],{},[91,2102],{"className":2103},[231,147]," y proporción ",[91,2106,2108,2126],{"className":2107},[94],[91,2109,2111],{"className":2110},[98],[100,2112,2113],{"xmlns":102},[104,2114,2115,2123],{},[107,2116,2117],{},[110,2118,2119,2121],{},[117,2120,980],{},[117,2122,583],{},[121,2124,2125],{"encoding":123},"\\frac{3}{5}",[91,2127,2129],{"className":2128,"ariaHidden":129},[128],[91,2130,2132,2135],{"className":2131},[133],[91,2133],{"className":2134,"style":347},[137],[91,2136,2138,2141,2200],{"className":2137},[142],[91,2139],{"className":2140},[146,147],[91,2142,2144],{"className":2143},[110],[91,2145,2147,2192],{"className":2146},[154,155],[91,2148,2150,2189],{"className":2149},[159],[91,2151,2153,2167,2175],{"className":2152,"style":366},[163],[91,2154,2155,2158],{"style":167},[91,2156],{"className":2157,"style":172},[171],[91,2159,2161],{"className":2160},[176,177,178,179],[91,2162,2164],{"className":2163},[142,179],[91,2165,583],{"className":2166},[142,179],[91,2168,2169,2172],{"style":188},[91,2170],{"className":2171,"style":172},[171],[91,2173],{"className":2174,"style":196},[195],[91,2176,2177,2180],{"style":199},[91,2178],{"className":2179,"style":172},[171],[91,2181,2183],{"className":2182},[176,177,178,179],[91,2184,2186],{"className":2185},[142,179],[91,2187,980],{"className":2188},[142,179],[91,2190,218],{"className":2191},[217],[91,2193,2195],{"className":2194},[159],[91,2196,2198],{"className":2197,"style":225},[163],[91,2199],{},[91,2201],{"className":2202},[231,147]," (3 caras de 5 lanzamientos).",[635,2205,2207],{"id":2206},"longitud-7","Longitud 7",[17,2209,2210,2211,2214],{},"El mismo razonamiento se aplica a secuencias de longitud 7 (4 caras, 3 cruces). Si las enumeramos todas — y créeme, lleva un rato — encontramos exactamente ",[33,2212,2213],{},"5 secuencias"," válidas.",[635,2216,2218],{"id":2217},"la-tabla-resumen","La tabla resumen",[2220,2221,2222,2301],"table",{},[2223,2224,2225],"thead",{},[2226,2227,2228,2292,2295,2298],"tr",{},[2229,2230,2232,2233,1090],"th",{"align":2231},"center","Longitud (",[91,2234,2236,2257],{"className":2235},[94],[91,2237,2239],{"className":2238},[98],[100,2240,2241],{"xmlns":102},[104,2242,2243,2254],{},[107,2244,2245,2247,2250,2252],{},[117,2246,458],{},[113,2248,2249],{},"n",[330,2251,1368],{},[117,2253,326],{},[121,2255,2256],{"encoding":123},"2n+1",[91,2258,2260,2283],{"className":2259,"ariaHidden":129},[128],[91,2261,2263,2266,2269,2272,2276,2280],{"className":2262},[133],[91,2264],{"className":2265,"style":1383},[137],[91,2267,458],{"className":2268},[142],[91,2270,2249],{"className":2271},[142,212],[91,2273],{"className":2274,"style":2275},[420],"margin-right:0.2222em;",[91,2277,1368],{"className":2278},[2279],"mbin",[91,2281],{"className":2282,"style":2275},[420],[91,2284,2286,2289],{"className":2285},[133],[91,2287],{"className":2288,"style":435},[137],[91,2290,326],{"className":2291},[142],[2229,2293,2294],{"align":2231},"Proporción de caras",[2229,2296,2297],{"align":2231},"Secuencias válidas",[2229,2299,2300],{"align":2231},"Probabilidad de cada una",[2302,2303,2304,2562,2819,3076,3336],"tbody",{},[2226,2305,2306,2361,2461,2463],{},[2307,2308,2309,2310,1090],"td",{"align":2231},"1 (",[91,2311,2313,2331],{"className":2312},[94],[91,2314,2316],{"className":2315},[98],[100,2317,2318],{"xmlns":102},[104,2319,2320,2328],{},[107,2321,2322,2324,2326],{},[113,2323,2249],{},[330,2325,332],{},[117,2327,571],{},[121,2329,2330],{"encoding":123},"n=0",[91,2332,2334,2352],{"className":2333,"ariaHidden":129},[128],[91,2335,2337,2340,2343,2346,2349],{"className":2336},[133],[91,2338],{"className":2339,"style":261},[137],[91,2341,2249],{"className":2342},[142,212],[91,2344],{"className":2345,"style":421},[420],[91,2347,332],{"className":2348},[425],[91,2350],{"className":2351,"style":421},[420],[91,2353,2355,2358],{"className":2354},[133],[91,2356],{"className":2357,"style":435},[137],[91,2359,571],{"className":2360},[142],[2307,2362,2363],{"align":2231},[91,2364,2366,2384],{"className":2365},[94],[91,2367,2369],{"className":2368},[98],[100,2370,2371],{"xmlns":102},[104,2372,2373,2381],{},[107,2374,2375],{},[110,2376,2377,2379],{},[117,2378,326],{},[117,2380,326],{},[121,2382,2383],{"encoding":123},"\\frac{1}{1}",[91,2385,2387],{"className":2386,"ariaHidden":129},[128],[91,2388,2390,2393],{"className":2389},[133],[91,2391],{"className":2392,"style":347},[137],[91,2394,2396,2399,2458],{"className":2395},[142],[91,2397],{"className":2398},[146,147],[91,2400,2402],{"className":2401},[110],[91,2403,2405,2450],{"className":2404},[154,155],[91,2406,2408,2447],{"className":2407},[159],[91,2409,2411,2425,2433],{"className":2410,"style":366},[163],[91,2412,2413,2416],{"style":167},[91,2414],{"className":2415,"style":172},[171],[91,2417,2419],{"className":2418},[176,177,178,179],[91,2420,2422],{"className":2421},[142,179],[91,2423,326],{"className":2424},[142,179],[91,2426,2427,2430],{"style":188},[91,2428],{"className":2429,"style":172},[171],[91,2431],{"className":2432,"style":196},[195],[91,2434,2435,2438],{"style":199},[91,2436],{"className":2437,"style":172},[171],[91,2439,2441],{"className":2440},[176,177,178,179],[91,2442,2444],{"className":2443},[142,179],[91,2445,326],{"className":2446},[142,179],[91,2448,218],{"className":2449},[217],[91,2451,2453],{"className":2452},[159],[91,2454,2456],{"className":2455,"style":225},[163],[91,2457],{},[91,2459],{"className":2460},[231,147],[2307,2462,326],{"align":2231},[2307,2464,2465],{"align":2231},[91,2466,2468,2485],{"className":2467},[94],[91,2469,2471],{"className":2470},[98],[100,2472,2473],{"xmlns":102},[104,2474,2475,2483],{},[107,2476,2477],{},[110,2478,2479,2481],{},[117,2480,326],{},[117,2482,458],{},[121,2484,461],{"encoding":123},[91,2486,2488],{"className":2487,"ariaHidden":129},[128],[91,2489,2491,2494],{"className":2490},[133],[91,2492],{"className":2493,"style":347},[137],[91,2495,2497,2500,2559],{"className":2496},[142],[91,2498],{"className":2499},[146,147],[91,2501,2503],{"className":2502},[110],[91,2504,2506,2551],{"className":2505},[154,155],[91,2507,2509,2548],{"className":2508},[159],[91,2510,2512,2526,2534],{"className":2511,"style":366},[163],[91,2513,2514,2517],{"style":167},[91,2515],{"className":2516,"style":172},[171],[91,2518,2520],{"className":2519},[176,177,178,179],[91,2521,2523],{"className":2522},[142,179],[91,2524,458],{"className":2525},[142,179],[91,2527,2528,2531],{"style":188},[91,2529],{"className":2530,"style":172},[171],[91,2532],{"className":2533,"style":196},[195],[91,2535,2536,2539],{"style":199},[91,2537],{"className":2538,"style":172},[171],[91,2540,2542],{"className":2541},[176,177,178,179],[91,2543,2545],{"className":2544},[142,179],[91,2546,326],{"className":2547},[142,179],[91,2549,218],{"className":2550},[217],[91,2552,2554],{"className":2553},[159],[91,2555,2557],{"className":2556,"style":225},[163],[91,2558],{},[91,2560],{"className":2561},[231,147],[2226,2563,2564,2618,2717,2719],{},[2307,2565,2566,2567,1090],{"align":2231},"3 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(",[91,2825,2827,2845],{"className":2826},[94],[91,2828,2830],{"className":2829},[98],[100,2831,2832],{"xmlns":102},[104,2833,2834,2842],{},[107,2835,2836,2838,2840],{},[113,2837,2249],{},[330,2839,332],{},[117,2841,458],{},[121,2843,2844],{"encoding":123},"n=2",[91,2846,2848,2866],{"className":2847,"ariaHidden":129},[128],[91,2849,2851,2854,2857,2860,2863],{"className":2850},[133],[91,2852],{"className":2853,"style":261},[137],[91,2855,2249],{"className":2856},[142,212],[91,2858],{"className":2859,"style":421},[420],[91,2861,332],{"className":2862},[425],[91,2864],{"className":2865,"style":421},[420],[91,2867,2869,2872],{"className":2868},[133],[91,2870],{"className":2871,"style":435},[137],[91,2873,458],{"className":2874},[142],[2307,2876,2877],{"align":2231},[91,2878,2880,2897],{"className":2879},[94],[91,2881,2883],{"className":2882},[98],[100,2884,2885],{"xmlns":102},[104,2886,2887,2895],{},[107,2888,2889],{},[110,2890,2891,2893],{},[117,2892,980],{},[117,2894,583],{},[121,2896,2125],{"encoding":123},[91,2898,2900],{"className":2899,"ariaHidden":129},[128],[91,2901,2903,2906],{"className":2902},[133],[91,2904],{"className":2905,"style":347},[137],[91,2907,2909,2912,2971],{"className":2908},[142],[91,2910],{"className":2911},[146,147],[91,2913,2915],{"className":2914},[110],[91,2916,2918,2963],{"className":2917},[154,155],[91,2919,2921,2960],{"className":2920},[159],[91,2922,2924,2938,2946],{"className":2923,"style":366},[163],[91,2925,2926,2929],{"style":167},[91,2927],{"className":2928,"style":172},[171],[91,2930,2932],{"className":2931},[176,177,178,179],[91,2933,2935],{"className":2934},[142,179],[91,2936,583],{"className":2937},[142,179],[91,2939,2940,2943],{"style":188},[91,2941],{"className":2942,"style":172},[171],[91,2944],{"className":2945,"style":196},[195],[91,2947,2948,2951],{"style":199},[91,2949],{"className":2950,"style":172},[171],[91,2952,2954],{"className":2953},[176,177,178,179],[91,2955,2957],{"className":2956},[142,179],[91,2958,980],{"className":2959},[142,179],[91,2961,218],{"className":2962},[217],[91,2964,2966],{"className":2965},[159],[91,2967,2969],{"className":2968,"style":225},[163],[91,2970],{},[91,2972],{"className":2973},[231,147],[2307,2975,458],{"align":2231},[2307,2977,2978],{"align":2231},[91,2979,2981,2999],{"className":2980},[94],[91,2982,2984],{"className":2983},[98],[100,2985,2986],{"xmlns":102},[104,2987,2988,2996],{},[107,2989,2990],{},[110,2991,2992,2994],{},[117,2993,326],{},[117,2995,1899],{},[121,2997,2998],{"encoding":123},"\\frac{1}{32}",[91,3000,3002],{"className":3001,"ariaHidden":129},[128],[91,3003,3005,3008],{"className":3004},[133],[91,3006],{"className":3007,"style":347},[137],[91,3009,3011,3014,3073],{"className":3010},[142],[91,3012],{"className":3013},[146,147],[91,3015,3017],{"className":3016},[110],[91,3018,3020,3065],{"className":3019},[154,155],[91,3021,3023,3062],{"className":3022},[159],[91,3024,3026,3040,3048],{"className":3025,"style":366},[163],[91,3027,3028,3031],{"style":167},[91,3029],{"className":3030,"style":172},[171],[91,3032,3034],{"className":3033},[176,177,178,179],[91,3035,3037],{"className":3036},[142,179],[91,3038,1899],{"className":3039},[142,179],[91,3041,3042,3045],{"style":188},[91,3043],{"className":3044,"style":172},[171],[91,3046],{"className":3047,"style":196},[195],[91,3049,3050,3053],{"style":199},[91,3051],{"className":3052,"style":172},[171],[91,3054,3056],{"className":3055},[176,177,178,179],[91,3057,3059],{"className":3058},[142,179],[91,3060,326],{"className":3061},[142,179],[91,3063,218],{"className":3064},[217],[91,3066,3068],{"className":3067},[159],[91,3069,3071],{"className":3070,"style":225},[163],[91,3072],{},[91,3074],{"className":3075},[231,147],[2226,3077,3078,3132,3233,3235],{},[2307,3079,3080,3081,1090],{"align":2231},"7 (",[91,3082,3084,3102],{"className":3083},[94],[91,3085,3087],{"className":3086},[98],[100,3088,3089],{"xmlns":102},[104,3090,3091,3099],{},[107,3092,3093,3095,3097],{},[113,3094,2249],{},[330,3096,332],{},[117,3098,980],{},[121,3100,3101],{"encoding":123},"n=3",[91,3103,3105,3123],{"className":3104,"ariaHidden":129},[128],[91,3106,3108,3111,3114,3117,3120],{"className":3107},[133],[91,3109],{"className":3110,"style":261},[137],[91,3112,2249],{"className":3113},[142,212],[91,3115],{"className":3116,"style":421},[420],[91,3118,332],{"className":3119},[425],[91,3121],{"className":3122,"style":421},[420],[91,3124,3126,3129],{"className":3125},[133],[91,3127],{"className":3128,"style":435},[137],[91,3130,980],{"className":3131},[142],[2307,3133,3134],{"align":2231},[91,3135,3137,3156],{"className":3136},[94],[91,3138,3140],{"className":3139},[98],[100,3141,3142],{"xmlns":102},[104,3143,3144,3153],{},[107,3145,3146],{},[110,3147,3148,3150],{},[117,3149,119],{},[117,3151,3152],{},"7",[121,3154,3155],{"encoding":123},"\\frac{4}{7}",[91,3157,3159],{"className":3158,"ariaHidden":129},[128],[91,3160,3162,3165],{"className":3161},[133],[91,3163],{"className":3164,"style":347},[137],[91,3166,3168,3171,3230],{"className":3167},[142],[91,3169],{"className":3170},[146,147],[91,3172,3174],{"className":3173},[110],[91,3175,3177,3222],{"className":3176},[154,155],[91,3178,3180,3219],{"className":3179},[159],[91,3181,3183,3197,3205],{"className":3182,"style":366},[163],[91,3184,3185,3188],{"style":167},[91,3186],{"className":3187,"style":172},[171],[91,3189,3191],{"className":3190},[176,177,178,179],[91,3192,3194],{"className":3193},[142,179],[91,3195,3152],{"className":3196},[142,179],[91,3198,3199,3202],{"style":188},[91,3200],{"className":3201,"style":172},[171],[91,3203],{"className":3204,"style":196},[195],[91,3206,3207,3210],{"style":199},[91,3208],{"className":3209,"style":172},[171],[91,3211,3213],{"className":3212},[176,177,178,179],[91,3214,3216],{"className":3215},[142,179],[91,3217,119],{"className":3218},[142,179],[91,3220,218],{"className":3221},[217],[91,3223,3225],{"className":3224},[159],[91,3226,3228],{"className":3227,"style":225},[163],[91,3229],{},[91,3231],{"className":3232},[231,147],[2307,3234,583],{"align":2231},[2307,3236,3237],{"align":2231},[91,3238,3240,3259],{"className":3239},[94],[91,3241,3243],{"className":3242},[98],[100,3244,3245],{"xmlns":102},[104,3246,3247,3256],{},[107,3248,3249],{},[110,3250,3251,3253],{},[117,3252,326],{},[117,3254,3255],{},"128",[121,3257,3258],{"encoding":123},"\\frac{1}{128}",[91,3260,3262],{"className":3261,"ariaHidden":129},[128],[91,3263,3265,3268],{"className":3264},[133],[91,3266],{"className":3267,"style":347},[137],[91,3269,3271,3274,3333],{"className":3270},[142],[91,3272],{"className":3273},[146,147],[91,3275,3277],{"className":3276},[110],[91,3278,3280,3325],{"className":3279},[154,155],[91,3281,3283,3322],{"className":3282},[159],[91,3284,3286,3300,3308],{"className":3285,"style":366},[163],[91,3287,3288,3291],{"style":167},[91,3289],{"className":3290,"style":172},[171],[91,3292,3294],{"className":3293},[176,177,178,179],[91,3295,3297],{"className":3296},[142,179],[91,3298,3255],{"className":3299},[142,179],[91,3301,3302,3305],{"style":188},[91,3303],{"className":3304,"style":172},[171],[91,3306],{"className":3307,"style":196},[195],[91,3309,3310,3313],{"style":199},[91,3311],{"className":3312,"style":172},[171],[91,3314,3316],{"className":3315},[176,177,178,179],[91,3317,3319],{"className":3318},[142,179],[91,3320,326],{"className":3321},[142,179],[91,3323,218],{"className":3324},[217],[91,3326,3328],{"className":3327},[159],[91,3329,3331],{"className":3330,"style":225},[163],[91,3332],{},[91,3334],{"className":3335},[231,147],[2226,3337,3338,3392,3493,3496],{},[2307,3339,3340,3341,1090],{"align":2231},"9 (",[91,3342,3344,3362],{"className":3343},[94],[91,3345,3347],{"className":3346},[98],[100,3348,3349],{"xmlns":102},[104,3350,3351,3359],{},[107,3352,3353,3355,3357],{},[113,3354,2249],{},[330,3356,332],{},[117,3358,119],{},[121,3360,3361],{"encoding":123},"n=4",[91,3363,3365,3383],{"className":3364,"ariaHidden":129},[128],[91,3366,3368,3371,3374,3377,3380],{"className":3367},[133],[91,3369],{"className":3370,"style":261},[137],[91,3372,2249],{"className":3373},[142,212],[91,3375],{"className":3376,"style":421},[420],[91,3378,332],{"className":3379},[425],[91,3381],{"className":3382,"style":421},[420],[91,3384,3386,3389],{"className":3385},[133],[91,3387],{"className":3388,"style":435},[137],[91,3390,119],{"className":3391},[142],[2307,3393,3394],{"align":2231},[91,3395,3397,3416],{"className":3396},[94],[91,3398,3400],{"className":3399},[98],[100,3401,3402],{"xmlns":102},[104,3403,3404,3413],{},[107,3405,3406],{},[110,3407,3408,3410],{},[117,3409,583],{},[117,3411,3412],{},"9",[121,3414,3415],{"encoding":123},"\\frac{5}{9}",[91,3417,3419],{"className":3418,"ariaHidden":129},[128],[91,3420,3422,3425],{"className":3421},[133],[91,3423],{"className":3424,"style":347},[137],[91,3426,3428,3431,3490],{"className":3427},[142],[91,3429],{"className":3430},[146,147],[91,3432,3434],{"className":3433},[110],[91,3435,3437,3482],{"className":3436},[154,155],[91,3438,3440,3479],{"className":3439},[159],[91,3441,3443,3457,3465],{"className":3442,"style":366},[163],[91,3444,3445,3448],{"style":167},[91,3446],{"className":3447,"style":172},[171],[91,3449,3451],{"className":3450},[176,177,178,179],[91,3452,3454],{"className":3453},[142,179],[91,3455,3412],{"className":3456},[142,179],[91,3458,3459,3462],{"style":188},[91,3460],{"className":3461,"style":172},[171],[91,3463],{"className":3464,"style":196},[195],[91,3466,3467,3470],{"style":199},[91,3468],{"className":3469,"style":172},[171],[91,3471,3473],{"className":3472},[176,177,178,179],[91,3474,3476],{"className":3475},[142,179],[91,3477,583],{"className":3478},[142,179],[91,3480,218],{"className":3481},[217],[91,3483,3485],{"className":3484},[159],[91,3486,3488],{"className":3487,"style":225},[163],[91,3489],{},[91,3491],{"className":3492},[231,147],[2307,3494,3495],{"align":2231},"14",[2307,3497,3498],{"align":2231},[91,3499,3501,3520],{"className":3500},[94],[91,3502,3504],{"className":3503},[98],[100,3505,3506],{"xmlns":102},[104,3507,3508,3517],{},[107,3509,3510],{},[110,3511,3512,3514],{},[117,3513,326],{},[117,3515,3516],{},"512",[121,3518,3519],{"encoding":123},"\\frac{1}{512}",[91,3521,3523],{"className":3522,"ariaHidden":129},[128],[91,3524,3526,3529],{"className":3525},[133],[91,3527],{"className":3528,"style":347},[137],[91,3530,3532,3535,3594],{"className":3531},[142],[91,3533],{"className":3534},[146,147],[91,3536,3538],{"className":3537},[110],[91,3539,3541,3586],{"className":3540},[154,155],[91,3542,3544,3583],{"className":3543},[159],[91,3545,3547,3561,3569],{"className":3546,"style":366},[163],[91,3548,3549,3552],{"style":167},[91,3550],{"className":3551,"style":172},[171],[91,3553,3555],{"className":3554},[176,177,178,179],[91,3556,3558],{"className":3557},[142,179],[91,3559,3516],{"className":3560},[142,179],[91,3562,3563,3566],{"style":188},[91,3564],{"className":3565,"style":172},[171],[91,3567],{"className":3568,"style":196},[195],[91,3570,3571,3574],{"style":199},[91,3572],{"className":3573,"style":172},[171],[91,3575,3577],{"className":3576},[176,177,178,179],[91,3578,3580],{"className":3579},[142,179],[91,3581,326],{"className":3582},[142,179],[91,3584,218],{"className":3585},[217],[91,3587,3589],{"className":3588},[159],[91,3590,3592],{"className":3591,"style":225},[163],[91,3593],{},[91,3595],{"className":3596},[231,147],[17,3598,3599],{},"Observa los patrones:",[3601,3602,3603,3662,3794],"ul",{},[287,3604,3605,3606,40],{},"La longitud siempre es impar: ",[91,3607,3609,3629],{"className":3608},[94],[91,3610,3612],{"className":3611},[98],[100,3613,3614],{"xmlns":102},[104,3615,3616,3626],{},[107,3617,3618,3620,3622,3624],{},[117,3619,458],{},[113,3621,2249],{},[330,3623,1368],{},[117,3625,326],{},[121,3627,3628],{"encoding":123},"2n + 1",[91,3630,3632,3653],{"className":3631,"ariaHidden":129},[128],[91,3633,3635,3638,3641,3644,3647,3650],{"className":3634},[133],[91,3636],{"className":3637,"style":1383},[137],[91,3639,458],{"className":3640},[142],[91,3642,2249],{"className":3643},[142,212],[91,3645],{"className":3646,"style":2275},[420],[91,3648,1368],{"className":3649},[2279],[91,3651],{"className":3652,"style":2275},[420],[91,3654,3656,3659],{"className":3655},[133],[91,3657],{"className":3658,"style":435},[137],[91,3660,326],{"className":3661},[142],[287,3663,3664,3665,40],{},"La proporción de caras es siempre ",[91,3666,3668,3700],{"className":3667},[94],[91,3669,3671],{"className":3670},[98],[100,3672,3673],{"xmlns":102},[104,3674,3675,3697],{},[107,3676,3677],{},[110,3678,3679,3687],{},[107,3680,3681,3683,3685],{},[113,3682,2249],{},[330,3684,1368],{},[117,3686,326],{},[107,3688,3689,3691,3693,3695],{},[117,3690,458],{},[113,3692,2249],{},[330,3694,1368],{},[117,3696,326],{},[121,3698,3699],{"encoding":123},"\\frac{n+1}{2n+1}",[91,3701,3703],{"className":3702,"ariaHidden":129},[128],[91,3704,3706,3710],{"className":3705},[133],[91,3707],{"className":3708,"style":3709},[137],"height:1.2484em;vertical-align:-0.4033em;",[91,3711,3713,3716,3791],{"className":3712},[142],[91,3714],{"className":3715},[146,147],[91,3717,3719],{"className":3718},[110],[91,3720,3722,3782],{"className":3721},[154,155],[91,3723,3725,3779],{"className":3724},[159],[91,3726,3728,3751,3759],{"className":3727,"style":366},[163],[91,3729,3730,3733],{"style":167},[91,3731],{"className":3732,"style":172},[171],[91,3734,3736],{"className":3735},[176,177,178,179],[91,3737,3739,3742,3745,3748],{"className":3738},[142,179],[91,3740,458],{"className":3741},[142,179],[91,3743,2249],{"className":3744},[142,212,179],[91,3746,1368],{"className":3747},[2279,179],[91,3749,326],{"className":3750},[142,179],[91,3752,3753,3756],{"style":188},[91,3754],{"className":3755,"style":172},[171],[91,3757],{"className":3758,"style":196},[195],[91,3760,3761,3764],{"style":199},[91,3762],{"className":3763,"style":172},[171],[91,3765,3767],{"className":3766},[176,177,178,179],[91,3768,3770,3773,3776],{"className":3769},[142,179],[91,3771,2249],{"className":3772},[142,212,179],[91,3774,1368],{"className":3775},[2279,179],[91,3777,326],{"className":3778},[142,179],[91,3780,218],{"className":3781},[217],[91,3783,3785],{"className":3784},[159],[91,3786,3789],{"className":3787,"style":3788},[163],"height:0.4033em;",[91,3790],{},[91,3792],{"className":3793},[231,147],[287,3795,3796,3797,3966,3967,4022],{},"La probabilidad de cada secuencia individual es ",[91,3798,3800,3836],{"className":3799},[94],[91,3801,3803],{"className":3802},[98],[100,3804,3805],{"xmlns":102},[104,3806,3807,3833],{},[107,3808,3809],{},[1075,3810,3811,3823],{},[107,3812,3813,3815,3821],{},[330,3814,1081],{"fence":129},[110,3816,3817,3819],{},[117,3818,326],{},[117,3820,458],{},[330,3822,1090],{"fence":129},[107,3824,3825,3827,3829,3831],{},[117,3826,458],{},[113,3828,2249],{},[330,3830,1368],{},[117,3832,326],{},[121,3834,3835],{"encoding":123},"\\left(\\frac{1}{2}\\right)^{2n+1}",[91,3837,3839],{"className":3838,"ariaHidden":129},[128],[91,3840,3842,3845],{"className":3841},[133],[91,3843],{"className":3844,"style":1114},[137],[91,3846,3848,3931],{"className":3847},[1118],[91,3849,3851,3857,3925],{"className":3850},[1118],[91,3852,3854],{"className":3853,"style":1126},[146,1125],[91,3855,1081],{"className":3856},[1130,1131],[91,3858,3860,3863,3922],{"className":3859},[142],[91,3861],{"className":3862},[146,147],[91,3864,3866],{"className":3865},[110],[91,3867,3869,3914],{"className":3868},[154,155],[91,3870,3872,3911],{"className":3871},[159],[91,3873,3875,3889,3897],{"className":3874,"style":366},[163],[91,3876,3877,3880],{"style":167},[91,3878],{"className":3879,"style":172},[171],[91,3881,3883],{"className":3882},[176,177,178,179],[91,3884,3886],{"className":3885},[142,179],[91,3887,458],{"className":3888},[142,179],[91,3890,3891,3894],{"style":188},[91,3892],{"className":3893,"style":172},[171],[91,3895],{"className":3896,"style":196},[195],[91,3898,3899,3902],{"style":199},[91,3900],{"className":3901,"style":172},[171],[91,3903,3905],{"className":3904},[176,177,178,179],[91,3906,3908],{"className":3907},[142,179],[91,3909,326],{"className":3910},[142,179],[91,3912,218],{"className":3913},[217],[91,3915,3917],{"className":3916},[159],[91,3918,3920],{"className":3919,"style":225},[163],[91,3921],{},[91,3923],{"className":3924},[231,147],[91,3926,3928],{"className":3927,"style":1126},[231,1125],[91,3929,1090],{"className":3930},[1130,1131],[91,3932,3934],{"className":3933},[1209],[91,3935,3937],{"className":3936},[154],[91,3938,3940],{"className":3939},[159],[91,3941,3943],{"className":3942,"style":1219},[163],[91,3944,3945,3948],{"style":1222},[91,3946],{"className":3947,"style":1226},[171],[91,3949,3951],{"className":3950},[176,177,178,179],[91,3952,3954,3957,3960,3963],{"className":3953},[142,179],[91,3955,458],{"className":3956},[142,179],[91,3958,2249],{"className":3959},[142,212,179],[91,3961,1368],{"className":3962},[2279,179],[91,3964,326],{"className":3965},[142,179]," — cada uno de los ",[91,3968,3970,3989],{"className":3969},[94],[91,3971,3973],{"className":3972},[98],[100,3974,3975],{"xmlns":102},[104,3976,3977,3987],{},[107,3978,3979,3981,3983,3985],{},[117,3980,458],{},[113,3982,2249],{},[330,3984,1368],{},[117,3986,326],{},[121,3988,2256],{"encoding":123},[91,3990,3992,4013],{"className":3991,"ariaHidden":129},[128],[91,3993,3995,3998,4001,4004,4007,4010],{"className":3994},[133],[91,3996],{"className":3997,"style":1383},[137],[91,3999,458],{"className":4000},[142],[91,4002,2249],{"className":4003},[142,212],[91,4005],{"className":4006,"style":2275},[420],[91,4008,1368],{"className":4009},[2279],[91,4011],{"className":4012,"style":2275},[420],[91,4014,4016,4019],{"className":4015},[133],[91,4017],{"className":4018,"style":435},[137],[91,4020,326],{"className":4021},[142]," lanzamientos tiene una probabilidad de un medio.",[17,4024,4025,4026],{},"Pero la columna que realmente importa es la de las secuencias válidas: ",[91,4027,4029,4064],{"className":4028},[94],[91,4030,4032],{"className":4031},[98],[100,4033,4034],{"xmlns":102},[104,4035,4036,4061],{},[107,4037,4038,4040,4042,4044,4046,4048,4050,4052,4054,4056,4058],{},[117,4039,326],{},[330,4041,579],{"separator":129},[117,4043,326],{},[330,4045,579],{"separator":129},[117,4047,458],{},[330,4049,579],{"separator":129},[117,4051,583],{},[330,4053,579],{"separator":129},[117,4055,3495],{},[330,4057,579],{"separator":129},[330,4059,4060],{},"…",[121,4062,4063],{"encoding":123},"1, 1, 2, 5, 14, \\ldots",[91,4065,4067],{"className":4066,"ariaHidden":129},[128],[91,4068,4070,4073,4076,4079,4082,4085,4088,4091,4094,4097,4100,4103,4106,4109,4112,4115,4118],{"className":4069},[133],[91,4071],{"className":4072,"style":567},[137],[91,4074,326],{"className":4075},[142],[91,4077,579],{"className":4078},[578],[91,4080],{"className":4081,"style":1504},[420],[91,4083,326],{"className":4084},[142],[91,4086,579],{"className":4087},[578],[91,4089],{"className":4090,"style":1504},[420],[91,4092,458],{"className":4093},[142],[91,4095,579],{"className":4096},[578],[91,4098],{"className":4099,"style":1504},[420],[91,4101,583],{"className":4102},[142],[91,4104,579],{"className":4105},[578],[91,4107],{"className":4108,"style":1504},[420],[91,4110,3495],{"className":4111},[142],[91,4113,579],{"className":4114},[578],[91,4116],{"className":4117,"style":1504},[420],[91,4119,4060],{"className":4120},[1118],[17,4122,4123],{},"¿Te suena?",[12,4125,4127],{"id":4126},"del-conteo-manual-a-los-números-de-catalan","Del conteo manual a los números de Catalan",[17,4129,4130,4131,4253,4254,4261,4262,4265],{},"Esos números — ",[91,4132,4134,4178],{"className":4133},[94],[91,4135,4137],{"className":4136},[98],[100,4138,4139],{"xmlns":102},[104,4140,4141,4175],{},[107,4142,4143,4145,4147,4149,4151,4153,4155,4157,4159,4161,4163,4166,4168,4171,4173],{},[117,4144,326],{},[330,4146,579],{"separator":129},[117,4148,326],{},[330,4150,579],{"separator":129},[117,4152,458],{},[330,4154,579],{"separator":129},[117,4156,583],{},[330,4158,579],{"separator":129},[117,4160,3495],{},[330,4162,579],{"separator":129},[117,4164,4165],{},"42",[330,4167,579],{"separator":129},[117,4169,4170],{},"132",[330,4172,579],{"separator":129},[330,4174,4060],{},[121,4176,4177],{"encoding":123},"1, 1, 2, 5, 14, 42, 132, \\ldots",[91,4179,4181],{"className":4180,"ariaHidden":129},[128],[91,4182,4184,4187,4190,4193,4196,4199,4202,4205,4208,4211,4214,4217,4220,4223,4226,4229,4232,4235,4238,4241,4244,4247,4250],{"className":4183},[133],[91,4185],{"className":4186,"style":567},[137],[91,4188,326],{"className":4189},[142],[91,4191,579],{"className":4192},[578],[91,4194],{"className":4195,"style":1504},[420],[91,4197,326],{"className":4198},[142],[91,4200,579],{"className":4201},[578],[91,4203],{"className":4204,"style":1504},[420],[91,4206,458],{"className":4207},[142],[91,4209,579],{"className":4210},[578],[91,4212],{"className":4213,"style":1504},[420],[91,4215,583],{"className":4216},[142],[91,4218,579],{"className":4219},[578],[91,4221],{"className":4222,"style":1504},[420],[91,4224,3495],{"className":4225},[142],[91,4227,579],{"className":4228},[578],[91,4230],{"className":4231,"style":1504},[420],[91,4233,4165],{"className":4234},[142],[91,4236,579],{"className":4237},[578],[91,4239],{"className":4240,"style":1504},[420],[91,4242,4170],{"className":4243},[142],[91,4245,579],{"className":4246},[578],[91,4248],{"className":4249,"style":1504},[420],[91,4251,4060],{"className":4252},[1118]," — tienen nombre propio. Son los ",[24,4255,4258],{"href":4256,"rel":4257},"https:\u002F\u002Fes.wikipedia.org\u002Fwiki\u002FN%C3%BAmeros_de_Catalan",[81],[33,4259,4260],{},"números de Catalan",", una de las secuencias más ubicuas de toda la combinatoria. (Y no, no son de origen ",[267,4263,4264],{},"Catalán",".)",[17,4267,4268,4269,4276,4277,4280,4281,4284,4285,4290,4291,4294],{},"Su historia es en sí misma una historia de conexiones inesperadas. El matemático mongol ",[24,4270,4273],{"href":4271,"rel":4272},"https:\u002F\u002Fes.wikipedia.org\u002Fwiki\u002FMinggatu",[81],[33,4274,4275],{},"Mingantu"," (c. 1692–1763), trabajando en China, fue la primera persona conocida en utilizarlos — alrededor de la década de 1730, en series trigonométricas incluidas en su libro ",[267,4278,4279],{},"Ge Yuan Mi Lu Jie Fa",". Dos décadas más tarde, en 1751, ",[33,4282,4283],{},"Leonhard Euler"," ",[24,4286,4289],{"href":4287,"rel":4288},"https:\u002F\u002Fes.wikipedia.org\u002Fwiki\u002FN%C3%BAmeros_de_Catalan#Historia",[81],"descubrió la misma secuencia de forma independiente"," en una carta a Goldbach, esta vez contando las formas de dividir un polígono en triángulos. Y casi un siglo después de Mingantu, el matemático franco-belga ",[33,4292,4293],{},"Eugène Charles Catalan"," (1814–1894) los encontró de nuevo en 1838, al contar las formas de colocar paréntesis en un producto de factores. La secuencia lleva su nombre — acuñado por John Riordan en el siglo XX — a pesar de que los números habían estado apareciendo, bajo diferentes formas, durante más de cien años antes que él.",[17,4296,4297],{},"La misma secuencia, descubierta de forma independiente en tres partes del mundo y a lo largo de tres siglos, cada vez en un contexto completamente distinto. Si ese patrón te suena familiar... sigue leyendo.",[17,4299,4300,4301,4329],{},"El ",[91,4302,4304,4317],{"className":4303},[94],[91,4305,4307],{"className":4306},[98],[100,4308,4309],{"xmlns":102},[104,4310,4311,4315],{},[107,4312,4313],{},[113,4314,2249],{},[121,4316,2249],{"encoding":123},[91,4318,4320],{"className":4319,"ariaHidden":129},[128],[91,4321,4323,4326],{"className":4322},[133],[91,4324],{"className":4325,"style":261},[137],[91,4327,2249],{"className":4328},[142,212],"-ésimo número de Catalan se define como:",[17,4331,4333],{"className":4332},[645],[91,4334,4336,4387],{"className":4335},[94],[91,4337,4339],{"className":4338},[98],[100,4340,4341],{"xmlns":102},[104,4342,4343,4384],{},[107,4344,4345,4353,4355,4367],{},[4346,4347,4348,4351],"msub",{},[113,4349,4350],{},"C",[113,4352,2249],{},[330,4354,332],{},[110,4356,4357,4359],{},[117,4358,326],{},[107,4360,4361,4363,4365],{},[113,4362,2249],{},[330,4364,1368],{},[117,4366,326],{},[107,4368,4369,4371,4382],{},[330,4370,1081],{"fence":129},[110,4372,4374,4380],{"linethickness":4373},"0px",[107,4375,4376,4378],{},[117,4377,458],{},[113,4379,2249],{},[113,4381,2249],{},[330,4383,1090],{"fence":129},[121,4385,4386],{"encoding":123},"C_n = \\frac{1}{n+1}\\binom{2n}{n}",[91,4388,4390,4450],{"className":4389,"ariaHidden":129},[128],[91,4391,4393,4397,4441,4444,4447],{"className":4392},[133],[91,4394],{"className":4395,"style":4396},[137],"height:0.8333em;vertical-align:-0.15em;",[91,4398,4400,4404],{"className":4399},[142],[91,4401,4350],{"className":4402,"style":4403},[142,212],"margin-right:0.0715em;",[91,4405,4407],{"className":4406},[1209],[91,4408,4410,4432],{"className":4409},[154,155],[91,4411,4413,4429],{"className":4412},[159],[91,4414,4417],{"className":4415,"style":4416},[163],"height:0.1514em;",[91,4418,4420,4423],{"style":4419},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[91,4421],{"className":4422,"style":1226},[171],[91,4424,4426],{"className":4425},[176,177,178,179],[91,4427,2249],{"className":4428},[142,212,179],[91,4430,218],{"className":4431},[217],[91,4433,4435],{"className":4434},[159],[91,4436,4439],{"className":4437,"style":4438},[163],"height:0.15em;",[91,4440],{},[91,4442],{"className":4443,"style":421},[420],[91,4445,332],{"className":4446},[425],[91,4448],{"className":4449,"style":421},[420],[91,4451,4453,4457,4531],{"className":4452},[133],[91,4454],{"className":4455,"style":4456},[137],"height:1.2984em;vertical-align:-0.4033em;",[91,4458,4460,4463,4528],{"className":4459},[142],[91,4461],{"className":4462},[146,147],[91,4464,4466],{"className":4465},[110],[91,4467,4469,4520],{"className":4468},[154,155],[91,4470,4472,4517],{"className":4471},[159],[91,4473,4475,4495,4503],{"className":4474,"style":366},[163],[91,4476,4477,4480],{"style":167},[91,4478],{"className":4479,"style":172},[171],[91,4481,4483],{"className":4482},[176,177,178,179],[91,4484,4486,4489,4492],{"className":4485},[142,179],[91,4487,2249],{"className":4488},[142,212,179],[91,4490,1368],{"className":4491},[2279,179],[91,4493,326],{"className":4494},[142,179],[91,4496,4497,4500],{"style":188},[91,4498],{"className":4499,"style":172},[171],[91,4501],{"className":4502,"style":196},[195],[91,4504,4505,4508],{"style":199},[91,4506],{"className":4507,"style":172},[171],[91,4509,4511],{"className":4510},[176,177,178,179],[91,4512,4514],{"className":4513},[142,179],[91,4515,326],{"className":4516},[142,179],[91,4518,218],{"className":4519},[217],[91,4521,4523],{"className":4522},[159],[91,4524,4526],{"className":4525,"style":3788},[163],[91,4527],{},[91,4529],{"className":4530},[231,147],[91,4532,4534,4540,4597],{"className":4533},[142],[91,4535,4537],{"className":4536,"style":1126},[146,1125],[91,4538,1081],{"className":4539},[1130,1131],[91,4541,4543],{"className":4542},[110],[91,4544,4546,4589],{"className":4545},[154,155],[91,4547,4549,4586],{"className":4548},[159],[91,4550,4553,4568],{"className":4551,"style":4552},[163],"height:0.8951em;",[91,4554,4556,4559],{"style":4555},"top:-2.355em;",[91,4557],{"className":4558,"style":1226},[171],[91,4560,4562],{"className":4561},[176,177,178,179],[91,4563,4565],{"className":4564},[142,179],[91,4566,2249],{"className":4567},[142,212,179],[91,4569,4571,4574],{"style":4570},"top:-3.144em;",[91,4572],{"className":4573,"style":1226},[171],[91,4575,4577],{"className":4576},[176,177,178,179],[91,4578,4580,4583],{"className":4579},[142,179],[91,4581,458],{"className":4582},[142,179],[91,4584,2249],{"className":4585},[142,212,179],[91,4587,218],{"className":4588},[217],[91,4590,4592],{"className":4591},[159],[91,4593,4595],{"className":4594,"style":225},[163],[91,4596],{},[91,4598,4600],{"className":4599,"style":1126},[231,1125],[91,4601,1090],{"className":4602},[1130,1131],[17,4604,4605,4606,4714,4715,4749,4750,4778,4779,4782,4783,4811,4812,4845,4846,4874,4875,4908,4909,40],{},"donde ",[91,4607,4609,4635],{"className":4608},[94],[91,4610,4612],{"className":4611},[98],[100,4613,4614],{"xmlns":102},[104,4615,4616,4632],{},[107,4617,4618,4620,4630],{},[330,4619,1081],{"fence":129},[110,4621,4622,4628],{"linethickness":4373},[107,4623,4624,4626],{},[117,4625,458],{},[113,4627,2249],{},[113,4629,2249],{},[330,4631,1090],{"fence":129},[121,4633,4634],{"encoding":123},"\\binom{2n}{n}",[91,4636,4638],{"className":4637,"ariaHidden":129},[128],[91,4639,4641,4645],{"className":4640},[133],[91,4642],{"className":4643,"style":4644},[137],"height:1.2451em;vertical-align:-0.35em;",[91,4646,4648,4654,4708],{"className":4647},[142],[91,4649,4651],{"className":4650,"style":1126},[146,1125],[91,4652,1081],{"className":4653},[1130,1131],[91,4655,4657],{"className":4656},[110],[91,4658,4660,4700],{"className":4659},[154,155],[91,4661,4663,4697],{"className":4662},[159],[91,4664,4666,4680],{"className":4665,"style":4552},[163],[91,4667,4668,4671],{"style":4555},[91,4669],{"className":4670,"style":1226},[171],[91,4672,4674],{"className":4673},[176,177,178,179],[91,4675,4677],{"className":4676},[142,179],[91,4678,2249],{"className":4679},[142,212,179],[91,4681,4682,4685],{"style":4570},[91,4683],{"className":4684,"style":1226},[171],[91,4686,4688],{"className":4687},[176,177,178,179],[91,4689,4691,4694],{"className":4690},[142,179],[91,4692,458],{"className":4693},[142,179],[91,4695,2249],{"className":4696},[142,212,179],[91,4698,218],{"className":4699},[217],[91,4701,4703],{"className":4702},[159],[91,4704,4706],{"className":4705,"style":225},[163],[91,4707],{},[91,4709,4711],{"className":4710,"style":1126},[231,1125],[91,4712,1090],{"className":4713},[1130,1131]," — se lee «",[91,4716,4718,4734],{"className":4717},[94],[91,4719,4721],{"className":4720},[98],[100,4722,4723],{"xmlns":102},[104,4724,4725,4731],{},[107,4726,4727,4729],{},[117,4728,458],{},[113,4730,2249],{},[121,4732,4733],{"encoding":123},"2n",[91,4735,4737],{"className":4736,"ariaHidden":129},[128],[91,4738,4740,4743,4746],{"className":4739},[133],[91,4741],{"className":4742,"style":435},[137],[91,4744,458],{"className":4745},[142],[91,4747,2249],{"className":4748},[142,212]," sobre ",[91,4751,4753,4766],{"className":4752},[94],[91,4754,4756],{"className":4755},[98],[100,4757,4758],{"xmlns":102},[104,4759,4760,4764],{},[107,4761,4762],{},[113,4763,2249],{},[121,4765,2249],{"encoding":123},[91,4767,4769],{"className":4768,"ariaHidden":129},[128],[91,4770,4772,4775],{"className":4771},[133],[91,4773],{"className":4774,"style":261},[137],[91,4776,2249],{"className":4777},[142,212],"» — es el ",[33,4780,4781],{},"coeficiente binomial",": el número de formas de elegir ",[91,4784,4786,4799],{"className":4785},[94],[91,4787,4789],{"className":4788},[98],[100,4790,4791],{"xmlns":102},[104,4792,4793,4797],{},[107,4794,4795],{},[113,4796,2249],{},[121,4798,2249],{"encoding":123},[91,4800,4802],{"className":4801,"ariaHidden":129},[128],[91,4803,4805,4808],{"className":4804},[133],[91,4806],{"className":4807,"style":261},[137],[91,4809,2249],{"className":4810},[142,212]," elementos de un conjunto de ",[91,4813,4815,4830],{"className":4814},[94],[91,4816,4818],{"className":4817},[98],[100,4819,4820],{"xmlns":102},[104,4821,4822,4828],{},[107,4823,4824,4826],{},[117,4825,458],{},[113,4827,2249],{},[121,4829,4733],{"encoding":123},[91,4831,4833],{"className":4832,"ariaHidden":129},[128],[91,4834,4836,4839,4842],{"className":4835},[133],[91,4837],{"className":4838,"style":435},[137],[91,4840,458],{"className":4841},[142],[91,4843,2249],{"className":4844},[142,212],". En la práctica, responde a preguntas como «¿de cuántas formas puedo distribuir ",[91,4847,4849,4862],{"className":4848},[94],[91,4850,4852],{"className":4851},[98],[100,4853,4854],{"xmlns":102},[104,4855,4856,4860],{},[107,4857,4858],{},[113,4859,2249],{},[121,4861,2249],{"encoding":123},[91,4863,4865],{"className":4864,"ariaHidden":129},[128],[91,4866,4868,4871],{"className":4867},[133],[91,4869],{"className":4870,"style":261},[137],[91,4872,2249],{"className":4873},[142,212]," caras en ",[91,4876,4878,4893],{"className":4877},[94],[91,4879,4881],{"className":4880},[98],[100,4882,4883],{"xmlns":102},[104,4884,4885,4891],{},[107,4886,4887,4889],{},[117,4888,458],{},[113,4890,2249],{},[121,4892,4733],{"encoding":123},[91,4894,4896],{"className":4895,"ariaHidden":129},[128],[91,4897,4899,4902,4905],{"className":4898},[133],[91,4900],{"className":4901,"style":435},[137],[91,4903,458],{"className":4904},[142],[91,4906,2249],{"className":4907},[142,212]," lanzamientos?» Exploramos esta notación en detalle en el ",[24,4910,4912],{"href":276,"rel":4911},[81],"artículo sobre el triángulo de Tartaglia",[17,4914,4915],{},"Comprobemos con nuestros datos:",[17,4917,4919],{"className":4918},[645],[91,4920,4922,4997],{"className":4921},[94],[91,4923,4925],{"className":4924},[98],[100,4926,4927],{"xmlns":102},[104,4928,4929,4994],{},[107,4930,4931,4937,4939,4945,4957,4959,4961,4964,4970,4972,4978,4990,4992],{},[4346,4932,4933,4935],{},[113,4934,4350],{},[117,4936,571],{},[330,4938,332],{},[110,4940,4941,4943],{},[117,4942,326],{},[117,4944,326],{},[107,4946,4947,4949,4955],{},[330,4948,1081],{"fence":129},[110,4950,4951,4953],{"linethickness":4373},[117,4952,571],{},[117,4954,571],{},[330,4956,1090],{"fence":129},[330,4958,332],{},[117,4960,326],{},[420,4962],{"width":4963},"2em",[4346,4965,4966,4968],{},[113,4967,4350],{},[117,4969,326],{},[330,4971,332],{},[110,4973,4974,4976],{},[117,4975,326],{},[117,4977,458],{},[107,4979,4980,4982,4988],{},[330,4981,1081],{"fence":129},[110,4983,4984,4986],{"linethickness":4373},[117,4985,458],{},[117,4987,326],{},[330,4989,1090],{"fence":129},[330,4991,332],{},[117,4993,326],{},[121,4995,4996],{"encoding":123},"C_0 = \\frac{1}{1}\\binom{0}{0} = 1 \\qquad C_1 = \\frac{1}{2}\\binom{2}{1} = 1",[91,4998,5000,5056,5205,5267,5416],{"className":4999,"ariaHidden":129},[128],[91,5001,5003,5006,5047,5050,5053],{"className":5002},[133],[91,5004],{"className":5005,"style":4396},[137],[91,5007,5009,5012],{"className":5008},[142],[91,5010,4350],{"className":5011,"style":4403},[142,212],[91,5013,5015],{"className":5014},[1209],[91,5016,5018,5039],{"className":5017},[154,155],[91,5019,5021,5036],{"className":5020},[159],[91,5022,5025],{"className":5023,"style":5024},[163],"height:0.3011em;",[91,5026,5027,5030],{"style":4419},[91,5028],{"className":5029,"style":1226},[171],[91,5031,5033],{"className":5032},[176,177,178,179],[91,5034,571],{"className":5035},[142,179],[91,5037,218],{"className":5038},[217],[91,5040,5042],{"className":5041},[159],[91,5043,5045],{"className":5044,"style":4438},[163],[91,5046],{},[91,5048],{"className":5049,"style":421},[420],[91,5051,332],{"className":5052},[425],[91,5054],{"className":5055,"style":421},[420],[91,5057,5059,5062,5130,5196,5199,5202],{"className":5058},[133],[91,5060],{"className":5061,"style":4644},[137],[91,5063,5065,5068,5127],{"className":5064},[142],[91,5066],{"className":5067},[146,147],[91,5069,5071],{"className":5070},[110],[91,5072,5074,5119],{"className":5073},[154,155],[91,5075,5077,5116],{"className":5076},[159],[91,5078,5080,5094,5102],{"className":5079,"style":366},[163],[91,5081,5082,5085],{"style":167},[91,5083],{"className":5084,"style":172},[171],[91,5086,5088],{"className":5087},[176,177,178,179],[91,5089,5091],{"className":5090},[142,179],[91,5092,326],{"className":5093},[142,179],[91,5095,5096,5099],{"style":188},[91,5097],{"className":5098,"style":172},[171],[91,5100],{"className":5101,"style":196},[195],[91,5103,5104,5107],{"style":199},[91,5105],{"className":5106,"style":172},[171],[91,5108,5110],{"className":5109},[176,177,178,179],[91,5111,5113],{"className":5112},[142,179],[91,5114,326],{"className":5115},[142,179],[91,5117,218],{"className":5118},[217],[91,5120,5122],{"className":5121},[159],[91,5123,5125],{"className":5124,"style":225},[163],[91,5126],{},[91,5128],{"className":5129},[231,147],[91,5131,5133,5139,5190],{"className":5132},[142],[91,5134,5136],{"className":5135,"style":1126},[146,1125],[91,5137,1081],{"className":5138},[1130,1131],[91,5140,5142],{"className":5141},[110],[91,5143,5145,5182],{"className":5144},[154,155],[91,5146,5148,5179],{"className":5147},[159],[91,5149,5151,5165],{"className":5150,"style":4552},[163],[91,5152,5153,5156],{"style":4555},[91,5154],{"className":5155,"style":1226},[171],[91,5157,5159],{"className":5158},[176,177,178,179],[91,5160,5162],{"className":5161},[142,179],[91,5163,571],{"className":5164},[142,179],[91,5166,5167,5170],{"style":4570},[91,5168],{"className":5169,"style":1226},[171],[91,5171,5173],{"className":5172},[176,177,178,179],[91,5174,5176],{"className":5175},[142,179],[91,5177,571],{"className":5178},[142,179],[91,5180,218],{"className":5181},[217],[91,5183,5185],{"className":5184},[159],[91,5186,5188],{"className":5187,"style":225},[163],[91,5189],{},[91,5191,5193],{"className":5192,"style":1126},[231,1125],[91,5194,1090],{"className":5195},[1130,1131],[91,5197],{"className":5198,"style":421},[420],[91,5200,332],{"className":5201},[425],[91,5203],{"className":5204,"style":421},[420],[91,5206,5208,5211,5214,5218,5258,5261,5264],{"className":5207},[133],[91,5209],{"className":5210,"style":4396},[137],[91,5212,326],{"className":5213},[142],[91,5215],{"className":5216,"style":5217},[420],"margin-right:2em;",[91,5219,5221,5224],{"className":5220},[142],[91,5222,4350],{"className":5223,"style":4403},[142,212],[91,5225,5227],{"className":5226},[1209],[91,5228,5230,5250],{"className":5229},[154,155],[91,5231,5233,5247],{"className":5232},[159],[91,5234,5236],{"className":5235,"style":5024},[163],[91,5237,5238,5241],{"style":4419},[91,5239],{"className":5240,"style":1226},[171],[91,5242,5244],{"className":5243},[176,177,178,179],[91,5245,326],{"className":5246},[142,179],[91,5248,218],{"className":5249},[217],[91,5251,5253],{"className":5252},[159],[91,5254,5256],{"className":5255,"style":4438},[163],[91,5257],{},[91,5259],{"className":5260,"style":421},[420],[91,5262,332],{"className":5263},[425],[91,5265],{"className":5266,"style":421},[420],[91,5268,5270,5273,5341,5407,5410,5413],{"className":5269},[133],[91,5271],{"className":5272,"style":4644},[137],[91,5274,5276,5279,5338],{"className":5275},[142],[91,5277],{"className":5278},[146,147],[91,5280,5282],{"className":5281},[110],[91,5283,5285,5330],{"className":5284},[154,155],[91,5286,5288,5327],{"className":5287},[159],[91,5289,5291,5305,5313],{"className":5290,"style":366},[163],[91,5292,5293,5296],{"style":167},[91,5294],{"className":5295,"style":172},[171],[91,5297,5299],{"className":5298},[176,177,178,179],[91,5300,5302],{"className":5301},[142,179],[91,5303,458],{"className":5304},[142,179],[91,5306,5307,5310],{"style":188},[91,5308],{"className":5309,"style":172},[171],[91,5311],{"className":5312,"style":196},[195],[91,5314,5315,5318],{"style":199},[91,5316],{"className":5317,"style":172},[171],[91,5319,5321],{"className":5320},[176,177,178,179],[91,5322,5324],{"className":5323},[142,179],[91,5325,326],{"className":5326},[142,179],[91,5328,218],{"className":5329},[217],[91,5331,5333],{"className":5332},[159],[91,5334,5336],{"className":5335,"style":225},[163],[91,5337],{},[91,5339],{"className":5340},[231,147],[91,5342,5344,5350,5401],{"className":5343},[142],[91,5345,5347],{"className":5346,"style":1126},[146,1125],[91,5348,1081],{"className":5349},[1130,1131],[91,5351,5353],{"className":5352},[110],[91,5354,5356,5393],{"className":5355},[154,155],[91,5357,5359,5390],{"className":5358},[159],[91,5360,5362,5376],{"className":5361,"style":4552},[163],[91,5363,5364,5367],{"style":4555},[91,5365],{"className":5366,"style":1226},[171],[91,5368,5370],{"className":5369},[176,177,178,179],[91,5371,5373],{"className":5372},[142,179],[91,5374,326],{"className":5375},[142,179],[91,5377,5378,5381],{"style":4570},[91,5379],{"className":5380,"style":1226},[171],[91,5382,5384],{"className":5383},[176,177,178,179],[91,5385,5387],{"className":5386},[142,179],[91,5388,458],{"className":5389},[142,179],[91,5391,218],{"className":5392},[217],[91,5394,5396],{"className":5395},[159],[91,5397,5399],{"className":5398,"style":225},[163],[91,5400],{},[91,5402,5404],{"className":5403,"style":1126},[231,1125],[91,5405,1090],{"className":5406},[1130,1131],[91,5408],{"className":5409,"style":421},[420],[91,5411,332],{"className":5412},[425],[91,5414],{"className":5415,"style":421},[420],[91,5417,5419,5422],{"className":5418},[133],[91,5420],{"className":5421,"style":435},[137],[91,5423,326],{"className":5424},[142],[17,5426,5428],{"className":5427},[645],[91,5429,5431,5506],{"className":5430},[94],[91,5432,5434],{"className":5433},[98],[100,5435,5436],{"xmlns":102},[104,5437,5438,5503],{},[107,5439,5440,5446,5448,5454,5466,5468,5470,5472,5478,5480,5486,5499,5501],{},[4346,5441,5442,5444],{},[113,5443,4350],{},[117,5445,458],{},[330,5447,332],{},[110,5449,5450,5452],{},[117,5451,326],{},[117,5453,980],{},[107,5455,5456,5458,5464],{},[330,5457,1081],{"fence":129},[110,5459,5460,5462],{"linethickness":4373},[117,5461,119],{},[117,5463,458],{},[330,5465,1090],{"fence":129},[330,5467,332],{},[117,5469,458],{},[420,5471],{"width":4963},[4346,5473,5474,5476],{},[113,5475,4350],{},[117,5477,980],{},[330,5479,332],{},[110,5481,5482,5484],{},[117,5483,326],{},[117,5485,119],{},[107,5487,5488,5490,5497],{},[330,5489,1081],{"fence":129},[110,5491,5492,5495],{"linethickness":4373},[117,5493,5494],{},"6",[117,5496,980],{},[330,5498,1090],{"fence":129},[330,5500,332],{},[117,5502,583],{},[121,5504,5505],{"encoding":123},"C_2 = \\frac{1}{3}\\binom{4}{2} = 2 \\qquad C_3 = \\frac{1}{4}\\binom{6}{3} = 5 ",[91,5507,5509,5564,5713,5774,5923],{"className":5508,"ariaHidden":129},[128],[91,5510,5512,5515,5555,5558,5561],{"className":5511},[133],[91,5513],{"className":5514,"style":4396},[137],[91,5516,5518,5521],{"className":5517},[142],[91,5519,4350],{"className":5520,"style":4403},[142,212],[91,5522,5524],{"className":5523},[1209],[91,5525,5527,5547],{"className":5526},[154,155],[91,5528,5530,5544],{"className":5529},[159],[91,5531,5533],{"className":5532,"style":5024},[163],[91,5534,5535,5538],{"style":4419},[91,5536],{"className":5537,"style":1226},[171],[91,5539,5541],{"className":5540},[176,177,178,179],[91,5542,458],{"className":5543},[142,179],[91,5545,218],{"className":5546},[217],[91,5548,5550],{"className":5549},[159],[91,5551,5553],{"className":5552,"style":4438},[163],[91,5554],{},[91,5556],{"className":5557,"style":421},[420],[91,5559,332],{"className":5560},[425],[91,5562],{"className":5563,"style":421},[420],[91,5565,5567,5570,5638,5704,5707,5710],{"className":5566},[133],[91,5568],{"className":5569,"style":4644},[137],[91,5571,5573,5576,5635],{"className":5572},[142],[91,5574],{"className":5575},[146,147],[91,5577,5579],{"className":5578},[110],[91,5580,5582,5627],{"className":5581},[154,155],[91,5583,5585,5624],{"className":5584},[159],[91,5586,5588,5602,5610],{"className":5587,"style":366},[163],[91,5589,5590,5593],{"style":167},[91,5591],{"className":5592,"style":172},[171],[91,5594,5596],{"className":5595},[176,177,178,179],[91,5597,5599],{"className":5598},[142,179],[91,5600,980],{"className":5601},[142,179],[91,5603,5604,5607],{"style":188},[91,5605],{"className":5606,"style":172},[171],[91,5608],{"className":5609,"style":196},[195],[91,5611,5612,5615],{"style":199},[91,5613],{"className":5614,"style":172},[171],[91,5616,5618],{"className":5617},[176,177,178,179],[91,5619,5621],{"className":5620},[142,179],[91,5622,326],{"className":5623},[142,179],[91,5625,218],{"className":5626},[217],[91,5628,5630],{"className":5629},[159],[91,5631,5633],{"className":5632,"style":225},[163],[91,5634],{},[91,5636],{"className":5637},[231,147],[91,5639,5641,5647,5698],{"className":5640},[142],[91,5642,5644],{"className":5643,"style":1126},[146,1125],[91,5645,1081],{"className":5646},[1130,1131],[91,5648,5650],{"className":5649},[110],[91,5651,5653,5690],{"className":5652},[154,155],[91,5654,5656,5687],{"className":5655},[159],[91,5657,5659,5673],{"className":5658,"style":4552},[163],[91,5660,5661,5664],{"style":4555},[91,5662],{"className":5663,"style":1226},[171],[91,5665,5667],{"className":5666},[176,177,178,179],[91,5668,5670],{"className":5669},[142,179],[91,5671,458],{"className":5672},[142,179],[91,5674,5675,5678],{"style":4570},[91,5676],{"className":5677,"style":1226},[171],[91,5679,5681],{"className":5680},[176,177,178,179],[91,5682,5684],{"className":5683},[142,179],[91,5685,119],{"className":5686},[142,179],[91,5688,218],{"className":5689},[217],[91,5691,5693],{"className":5692},[159],[91,5694,5696],{"className":5695,"style":225},[163],[91,5697],{},[91,5699,5701],{"className":5700,"style":1126},[231,1125],[91,5702,1090],{"className":5703},[1130,1131],[91,5705],{"className":5706,"style":421},[420],[91,5708,332],{"className":5709},[425],[91,5711],{"className":5712,"style":421},[420],[91,5714,5716,5719,5722,5725,5765,5768,5771],{"className":5715},[133],[91,5717],{"className":5718,"style":4396},[137],[91,5720,458],{"className":5721},[142],[91,5723],{"className":5724,"style":5217},[420],[91,5726,5728,5731],{"className":5727},[142],[91,5729,4350],{"className":5730,"style":4403},[142,212],[91,5732,5734],{"className":5733},[1209],[91,5735,5737,5757],{"className":5736},[154,155],[91,5738,5740,5754],{"className":5739},[159],[91,5741,5743],{"className":5742,"style":5024},[163],[91,5744,5745,5748],{"style":4419},[91,5746],{"className":5747,"style":1226},[171],[91,5749,5751],{"className":5750},[176,177,178,179],[91,5752,980],{"className":5753},[142,179],[91,5755,218],{"className":5756},[217],[91,5758,5760],{"className":5759},[159],[91,5761,5763],{"className":5762,"style":4438},[163],[91,5764],{},[91,5766],{"className":5767,"style":421},[420],[91,5769,332],{"className":5770},[425],[91,5772],{"className":5773,"style":421},[420],[91,5775,5777,5780,5848,5914,5917,5920],{"className":5776},[133],[91,5778],{"className":5779,"style":4644},[137],[91,5781,5783,5786,5845],{"className":5782},[142],[91,5784],{"className":5785},[146,147],[91,5787,5789],{"className":5788},[110],[91,5790,5792,5837],{"className":5791},[154,155],[91,5793,5795,5834],{"className":5794},[159],[91,5796,5798,5812,5820],{"className":5797,"style":366},[163],[91,5799,5800,5803],{"style":167},[91,5801],{"className":5802,"style":172},[171],[91,5804,5806],{"className":5805},[176,177,178,179],[91,5807,5809],{"className":5808},[142,179],[91,5810,119],{"className":5811},[142,179],[91,5813,5814,5817],{"style":188},[91,5815],{"className":5816,"style":172},[171],[91,5818],{"className":5819,"style":196},[195],[91,5821,5822,5825],{"style":199},[91,5823],{"className":5824,"style":172},[171],[91,5826,5828],{"className":5827},[176,177,178,179],[91,5829,5831],{"className":5830},[142,179],[91,5832,326],{"className":5833},[142,179],[91,5835,218],{"className":5836},[217],[91,5838,5840],{"className":5839},[159],[91,5841,5843],{"className":5842,"style":225},[163],[91,5844],{},[91,5846],{"className":5847},[231,147],[91,5849,5851,5857,5908],{"className":5850},[142],[91,5852,5854],{"className":5853,"style":1126},[146,1125],[91,5855,1081],{"className":5856},[1130,1131],[91,5858,5860],{"className":5859},[110],[91,5861,5863,5900],{"className":5862},[154,155],[91,5864,5866,5897],{"className":5865},[159],[91,5867,5869,5883],{"className":5868,"style":4552},[163],[91,5870,5871,5874],{"style":4555},[91,5872],{"className":5873,"style":1226},[171],[91,5875,5877],{"className":5876},[176,177,178,179],[91,5878,5880],{"className":5879},[142,179],[91,5881,980],{"className":5882},[142,179],[91,5884,5885,5888],{"style":4570},[91,5886],{"className":5887,"style":1226},[171],[91,5889,5891],{"className":5890},[176,177,178,179],[91,5892,5894],{"className":5893},[142,179],[91,5895,5494],{"className":5896},[142,179],[91,5898,218],{"className":5899},[217],[91,5901,5903],{"className":5902},[159],[91,5904,5906],{"className":5905,"style":225},[163],[91,5907],{},[91,5909,5911],{"className":5910,"style":1126},[231,1125],[91,5912,1090],{"className":5913},[1130,1131],[91,5915],{"className":5916,"style":421},[420],[91,5918,332],{"className":5919},[425],[91,5921],{"className":5922,"style":421},[420],[91,5924,5926,5929],{"className":5925},[133],[91,5927],{"className":5928,"style":435},[137],[91,5930,583],{"className":5931},[142],[17,5933,5935],{"className":5934},[645],[91,5936,5938,5980],{"className":5937},[94],[91,5939,5941],{"className":5940},[98],[100,5942,5943],{"xmlns":102},[104,5944,5945,5977],{},[107,5946,5947,5953,5955,5961,5973,5975],{},[4346,5948,5949,5951],{},[113,5950,4350],{},[117,5952,119],{},[330,5954,332],{},[110,5956,5957,5959],{},[117,5958,326],{},[117,5960,583],{},[107,5962,5963,5965,5971],{},[330,5964,1081],{"fence":129},[110,5966,5967,5969],{"linethickness":4373},[117,5968,1101],{},[117,5970,119],{},[330,5972,1090],{"fence":129},[330,5974,332],{},[117,5976,3495],{},[121,5978,5979],{"encoding":123},"C_4 = \\frac{1}{5}\\binom{8}{4} = 14",[91,5981,5983,6038,6187],{"className":5982,"ariaHidden":129},[128],[91,5984,5986,5989,6029,6032,6035],{"className":5985},[133],[91,5987],{"className":5988,"style":4396},[137],[91,5990,5992,5995],{"className":5991},[142],[91,5993,4350],{"className":5994,"style":4403},[142,212],[91,5996,5998],{"className":5997},[1209],[91,5999,6001,6021],{"className":6000},[154,155],[91,6002,6004,6018],{"className":6003},[159],[91,6005,6007],{"className":6006,"style":5024},[163],[91,6008,6009,6012],{"style":4419},[91,6010],{"className":6011,"style":1226},[171],[91,6013,6015],{"className":6014},[176,177,178,179],[91,6016,119],{"className":6017},[142,179],[91,6019,218],{"className":6020},[217],[91,6022,6024],{"className":6023},[159],[91,6025,6027],{"className":6026,"style":4438},[163],[91,6028],{},[91,6030],{"className":6031,"style":421},[420],[91,6033,332],{"className":6034},[425],[91,6036],{"className":6037,"style":421},[420],[91,6039,6041,6044,6112,6178,6181,6184],{"className":6040},[133],[91,6042],{"className":6043,"style":4644},[137],[91,6045,6047,6050,6109],{"className":6046},[142],[91,6048],{"className":6049},[146,147],[91,6051,6053],{"className":6052},[110],[91,6054,6056,6101],{"className":6055},[154,155],[91,6057,6059,6098],{"className":6058},[159],[91,6060,6062,6076,6084],{"className":6061,"style":366},[163],[91,6063,6064,6067],{"style":167},[91,6065],{"className":6066,"style":172},[171],[91,6068,6070],{"className":6069},[176,177,178,179],[91,6071,6073],{"className":6072},[142,179],[91,6074,583],{"className":6075},[142,179],[91,6077,6078,6081],{"style":188},[91,6079],{"className":6080,"style":172},[171],[91,6082],{"className":6083,"style":196},[195],[91,6085,6086,6089],{"style":199},[91,6087],{"className":6088,"style":172},[171],[91,6090,6092],{"className":6091},[176,177,178,179],[91,6093,6095],{"className":6094},[142,179],[91,6096,326],{"className":6097},[142,179],[91,6099,218],{"className":6100},[217],[91,6102,6104],{"className":6103},[159],[91,6105,6107],{"className":6106,"style":225},[163],[91,6108],{},[91,6110],{"className":6111},[231,147],[91,6113,6115,6121,6172],{"className":6114},[142],[91,6116,6118],{"className":6117,"style":1126},[146,1125],[91,6119,1081],{"className":6120},[1130,1131],[91,6122,6124],{"className":6123},[110],[91,6125,6127,6164],{"className":6126},[154,155],[91,6128,6130,6161],{"className":6129},[159],[91,6131,6133,6147],{"className":6132,"style":4552},[163],[91,6134,6135,6138],{"style":4555},[91,6136],{"className":6137,"style":1226},[171],[91,6139,6141],{"className":6140},[176,177,178,179],[91,6142,6144],{"className":6143},[142,179],[91,6145,119],{"className":6146},[142,179],[91,6148,6149,6152],{"style":4570},[91,6150],{"className":6151,"style":1226},[171],[91,6153,6155],{"className":6154},[176,177,178,179],[91,6156,6158],{"className":6157},[142,179],[91,6159,1101],{"className":6160},[142,179],[91,6162,218],{"className":6163},[217],[91,6165,6167],{"className":6166},[159],[91,6168,6170],{"className":6169,"style":225},[163],[91,6171],{},[91,6173,6175],{"className":6174,"style":1126},[231,1125],[91,6176,1090],{"className":6177},[1130,1131],[91,6179],{"className":6180,"style":421},[420],[91,6182,332],{"className":6183},[425],[91,6185],{"className":6186,"style":421},[420],[91,6188,6190,6193],{"className":6189},[133],[91,6191],{"className":6192,"style":435},[137],[91,6194,3495],{"className":6195},[142],[17,6197,6198,6199,6202],{},"Exacto. Los números de Catalan cuentan precisamente cuántas secuencias de lanzamientos terminan ",[33,6200,6201],{},"por primera vez"," con más caras que cruces en cada longitud posible.",[12,6204,6206],{"id":6205},"la-conexión-con-el-triángulo-de-tartaglia","La conexión con el triángulo de Tartaglia",[17,6208,6209,6210,6213,6214,6269,6270,6399,6400,6404,6405,40],{},"Hay otra forma de ver por qué aparecen los números de Catalan aquí. Pensemos en cada secuencia de lanzamientos como un ",[33,6211,6212],{},"camino"," en una retícula: cara sube un paso, cruz baja un paso. El número total de caminos de longitud ",[91,6215,6217,6236],{"className":6216},[94],[91,6218,6220],{"className":6219},[98],[100,6221,6222],{"xmlns":102},[104,6223,6224,6234],{},[107,6225,6226,6228,6230,6232],{},[117,6227,458],{},[113,6229,2249],{},[330,6231,1368],{},[117,6233,326],{},[121,6235,2256],{"encoding":123},[91,6237,6239,6260],{"className":6238,"ariaHidden":129},[128],[91,6240,6242,6245,6248,6251,6254,6257],{"className":6241},[133],[91,6243],{"className":6244,"style":1383},[137],[91,6246,458],{"className":6247},[142],[91,6249,2249],{"className":6250},[142,212],[91,6252],{"className":6253,"style":2275},[420],[91,6255,1368],{"className":6256},[2279],[91,6258],{"className":6259,"style":2275},[420],[91,6261,6263,6266],{"className":6262},[133],[91,6264],{"className":6265,"style":435},[137],[91,6267,326],{"className":6268},[142]," que terminan un paso por encima de cero es el coeficiente binomial ",[91,6271,6273,6309],{"className":6272},[94],[91,6274,6276],{"className":6275},[98],[100,6277,6278],{"xmlns":102},[104,6279,6280,6306],{},[107,6281,6282,6284,6304],{},[330,6283,1081],{"fence":129},[110,6285,6286,6296],{"linethickness":4373},[107,6287,6288,6290,6292,6294],{},[117,6289,458],{},[113,6291,2249],{},[330,6293,1368],{},[117,6295,326],{},[107,6297,6298,6300,6302],{},[113,6299,2249],{},[330,6301,1368],{},[117,6303,326],{},[330,6305,1090],{"fence":129},[121,6307,6308],{"encoding":123},"\\binom{2n+1}{n+1}",[91,6310,6312],{"className":6311,"ariaHidden":129},[128],[91,6313,6315,6318],{"className":6314},[133],[91,6316],{"className":6317,"style":4456},[137],[91,6319,6321,6327,6393],{"className":6320},[142],[91,6322,6324],{"className":6323,"style":1126},[146,1125],[91,6325,1081],{"className":6326},[1130,1131],[91,6328,6330],{"className":6329},[110],[91,6331,6333,6385],{"className":6332},[154,155],[91,6334,6336,6382],{"className":6335},[159],[91,6337,6339,6359],{"className":6338,"style":4552},[163],[91,6340,6341,6344],{"style":4555},[91,6342],{"className":6343,"style":1226},[171],[91,6345,6347],{"className":6346},[176,177,178,179],[91,6348,6350,6353,6356],{"className":6349},[142,179],[91,6351,2249],{"className":6352},[142,212,179],[91,6354,1368],{"className":6355},[2279,179],[91,6357,326],{"className":6358},[142,179],[91,6360,6361,6364],{"style":4570},[91,6362],{"className":6363,"style":1226},[171],[91,6365,6367],{"className":6366},[176,177,178,179],[91,6368,6370,6373,6376,6379],{"className":6369},[142,179],[91,6371,458],{"className":6372},[142,179],[91,6374,2249],{"className":6375},[142,212,179],[91,6377,1368],{"className":6378},[2279,179],[91,6380,326],{"className":6381},[142,179],[91,6383,218],{"className":6384},[217],[91,6386,6388],{"className":6387},[159],[91,6389,6391],{"className":6390,"style":3788},[163],[91,6392],{},[91,6394,6396],{"className":6395,"style":1126},[231,1125],[91,6397,1090],{"className":6398},[1130,1131]," — una entrada del triángulo de Tartaglia. Los números de Catalan cuentan sólo los caminos que nunca cruzan por encima de cero antes del paso final. Son, literalmente, una selección filtrada dentro del triángulo — la misma herramienta que usamos en el ",[24,6401,6403],{"href":276,"rel":6402},[81],"artículo anterior"," para contar combinaciones de monedas, ahora con una restricción adicional. Si quieres explorar esta conexión en más detalle, Matt Parker la explica de forma brillante en ",[24,6406,6408],{"href":86,"rel":6407},[81],"su vídeo",[12,6410,6412],{"id":6411},"la-serie-infinita","La serie infinita",[17,6414,6415,6416,6471],{},"Volvamos a nuestra tabla. La contribución de las secuencias de longitud ",[91,6417,6419,6438],{"className":6418},[94],[91,6420,6422],{"className":6421},[98],[100,6423,6424],{"xmlns":102},[104,6425,6426,6436],{},[107,6427,6428,6430,6432,6434],{},[117,6429,458],{},[113,6431,2249],{},[330,6433,1368],{},[117,6435,326],{},[121,6437,2256],{"encoding":123},[91,6439,6441,6462],{"className":6440,"ariaHidden":129},[128],[91,6442,6444,6447,6450,6453,6456,6459],{"className":6443},[133],[91,6445],{"className":6446,"style":1383},[137],[91,6448,458],{"className":6449},[142],[91,6451,2249],{"className":6452},[142,212],[91,6454],{"className":6455,"style":2275},[420],[91,6457,1368],{"className":6458},[2279],[91,6460],{"className":6461,"style":2275},[420],[91,6463,6465,6468],{"className":6464},[133],[91,6466],{"className":6467,"style":435},[137],[91,6469,326],{"className":6470},[142]," al promedio total 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= \\sum_{n=0}^{\\infty} \\frac{C_n}{2^{2n+1}} \\cdot \\frac{n+1}{2n+1}",[91,7421,7423,7444,7686],{"className":7422,"ariaHidden":129},[128],[91,7424,7426,7429,7435,7438,7441],{"className":7425},[133],[91,7427],{"className":7428,"style":7078},[137],[91,7430,7432],{"className":7431},[142,678],[91,7433,7351],{"className":7434},[142],[91,7436],{"className":7437,"style":421},[420],[91,7439,332],{"className":7440},[425],[91,7442],{"className":7443,"style":421},[420],[91,7445,7447,7451,7522,7525,7677,7680,7683],{"className":7446},[133],[91,7448],{"className":7449,"style":7450},[137],"height:1.2334em;vertical-align:-0.345em;",[91,7452,7455,7461],{"className":7453},[7454],"mop",[91,7456,7359],{"className":7457,"style":7460},[7454,7458,7459],"op-symbol","small-op","position:relative;top:0em;",[91,7462,7464],{"className":7463},[1209],[91,7465,7467,7513],{"className":7466},[154,155],[91,7468,7470,7510],{"className":7469},[159],[91,7471,7474,7495],{"className":7472,"style":7473},[163],"height:0.8043em;",[91,7475,7477,7480],{"style":7476},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[91,7478],{"className":7479,"style":1226},[171],[91,7481,7483],{"className":7482},[176,177,178,179],[91,7484,7486,7489,7492],{"className":7485},[142,179],[91,7487,2249],{"className":7488},[142,212,179],[91,7490,332],{"className":7491},[425,179],[91,7493,571],{"className":7494},[142,179],[91,7496,7498,7501],{"style":7497},"top:-3.2029em;margin-right:0.05em;",[91,7499],{"className":7500,"style":1226},[171],[91,7502,7504],{"className":7503},[176,177,178,179],[91,7505,7507],{"className":7506},[142,179],[91,7508,7371],{"className":7509},[142,179],[91,7511,218],{"className":7512},[217],[91,7514,7516],{"className":7515},[159],[91,7517,7520],{"className":7518,"style":7519},[163],"height:0.2997em;",[91,7521],{},[91,7523],{"className":7524,"style":1504},[420],[91,7526,7528,7531,7674],{"className":7527},[142],[91,7529],{"className":7530},[146,147],[91,7532,7534],{"className":7533},[110],[91,7535,7537,7666],{"className":7536},[154,155],[91,7538,7540,7663],{"className":7539},[159],[91,7541,7544,7600,7608],{"className":7542,"style":7543},[163],"height:0.8884em;",[91,7545,7546,7549],{"style":167},[91,7547],{"className":7548,"style":172},[171],[91,7550,7552],{"className":7551},[176,177,178,179],[91,7553,7555],{"className":7554},[142,179],[91,7556,7558,7561],{"className":7557},[142,179],[91,7559,458],{"className":7560},[142,179],[91,7562,7564],{"className":7563},[1209],[91,7565,7567],{"className":7566},[154],[91,7568,7570],{"className":7569},[159],[91,7571,7574],{"className":7572,"style":7573},[163],"height:0.7463em;",[91,7575,7577,7581],{"style":7576},"top:-2.786em;margin-right:0.0714em;",[91,7578],{"className":7579,"style":7580},[171],"height:2.5em;",[91,7582,7585],{"className":7583},[176,7584,1131,179],"reset-size3",[91,7586,7588,7591,7594,7597],{"className":7587},[142,179],[91,7589,458],{"className":7590},[142,179],[91,7592,2249],{"className":7593},[142,212,179],[91,7595,1368],{"className":7596},[2279,179],[91,7598,326],{"className":7599},[142,179],[91,7601,7602,7605],{"style":188},[91,7603],{"className":7604,"style":172},[171],[91,7606],{"className":7607,"style":196},[195],[91,7609,7611,7614],{"style":7610},"top:-3.4101em;",[91,7612],{"className":7613,"style":172},[171],[91,7615,7617],{"className":7616},[176,177,178,179],[91,7618,7620],{"className":7619},[142,179],[91,7621,7623,7626],{"className":7622},[142,179],[91,7624,4350],{"className":7625,"style":4403},[142,212,179],[91,7627,7629],{"className":7628},[1209],[91,7630,7632,7654],{"className":7631},[154,155],[91,7633,7635,7651],{"className":7634},[159],[91,7636,7639],{"className":7637,"style":7638},[163],"height:0.1645em;",[91,7640,7642,7645],{"style":7641},"top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;",[91,7643],{"className":7644,"style":7580},[171],[91,7646,7648],{"className":7647},[176,7584,1131,179],[91,7649,2249],{"className":7650},[142,212,179],[91,7652,218],{"className":7653},[217],[91,7655,7657],{"className":7656},[159],[91,7658,7661],{"className":7659,"style":7660},[163],"height:0.143em;",[91,7662],{},[91,7664,218],{"className":7665},[217],[91,7667,7669],{"className":7668},[159],[91,7670,7672],{"className":7671,"style":225},[163],[91,7673],{},[91,7675],{"className":7676},[231,147],[91,7678],{"className":7679,"style":2275},[420],[91,7681,7396],{"className":7682},[2279],[91,7684],{"className":7685,"style":2275},[420],[91,7687,7689,7692],{"className":7688},[133],[91,7690],{"className":7691,"style":3709},[137],[91,7693,7695,7698,7772],{"className":7694},[142],[91,7696],{"className":7697},[146,147],[91,7699,7701],{"className":7700},[110],[91,7702,7704,7764],{"className":7703},[154,155],[91,7705,7707,7761],{"className":7706},[159],[91,7708,7710,7733,7741],{"className":7709,"style":366},[163],[91,7711,7712,7715],{"style":167},[91,7713],{"className":7714,"style":172},[171],[91,7716,7718],{"className":7717},[176,177,178,179],[91,7719,7721,7724,7727,7730],{"className":7720},[142,179],[91,7722,458],{"className":7723},[142,179],[91,7725,2249],{"className":7726},[142,212,179],[91,7728,1368],{"className":7729},[2279,179],[91,7731,326],{"className":7732},[142,179],[91,7734,7735,7738],{"style":188},[91,7736],{"className":7737,"style":172},[171],[91,7739],{"className":7740,"style":196},[195],[91,7742,7743,7746],{"style":199},[91,7744],{"className":7745,"style":172},[171],[91,7747,7749],{"className":7748},[176,177,178,179],[91,7750,7752,7755,7758],{"className":7751},[142,179],[91,7753,2249],{"className":7754},[142,212,179],[91,7756,1368],{"className":7757},[2279,179],[91,7759,326],{"className":7760},[142,179],[91,7762,218],{"className":7763},[217],[91,7765,7767],{"className":7766},[159],[91,7768,7770],{"className":7769,"style":3788},[163],[91,7771],{},[91,7773],{"className":7774},[231,147],[17,7776,7777],{},"Calculemos los primeros términos:",[2220,7779,7780,7821],{},[2223,7781,7782],{},[2226,7783,7784,7814,7818],{},[2229,7785,7786],{"align":2231},[91,7787,7789,7802],{"className":7788},[94],[91,7790,7792],{"className":7791},[98],[100,7793,7794],{"xmlns":102},[104,7795,7796,7800],{},[107,7797,7798],{},[113,7799,2249],{},[121,7801,2249],{"encoding":123},[91,7803,7805],{"className":7804,"ariaHidden":129},[128],[91,7806,7808,7811],{"className":7807},[133],[91,7809],{"className":7810,"style":261},[137],[91,7812,2249],{"className":7813},[142,212],[2229,7815,7817],{"align":7816},"left","Contribución",[2229,7819,7820],{"align":7816},"Acumulado",[2302,7822,7823,8044,8337,8629,8921],{},[2226,7824,7825,7827,8002],{},[2307,7826,571],{"align":2231},[2307,7828,7829],{"align":7816},[91,7830,7832,7862],{"className":7831},[94],[91,7833,7835],{"className":7834},[98],[100,7836,7837],{"xmlns":102},[104,7838,7839,7859],{},[107,7840,7841,7847,7849,7851,7853,7855,7857],{},[110,7842,7843,7845],{},[117,7844,326],{},[117,7846,458],{},[330,7848,6524],{},[117,7850,326],{},[330,7852,6524],{},[117,7854,326],{},[330,7856,332],{},[117,7858,554],{},[121,7860,7861],{"encoding":123},"\\frac{1}{2} \\times 1 \\times 1 = 0{,}5",[91,7863,7865,7948,7966,7984],{"className":7864,"ariaHidden":129},[128],[91,7866,7868,7871,7939,7942,7945],{"className":7867},[133],[91,7869],{"className":7870,"style":347},[137],[91,7872,7874,7877,7936],{"className":7873},[142],[91,7875],{"className":7876},[146,147],[91,7878,7880],{"className":7879},[110],[91,7881,7883,7928],{"className":7882},[154,155],[91,7884,7886,7925],{"className":7885},[159],[91,7887,7889,7903,7911],{"className":7888,"style":366},[163],[91,7890,7891,7894],{"style":167},[91,7892],{"className":7893,"style":172},[171],[91,7895,7897],{"className":7896},[176,177,178,179],[91,7898,7900],{"className":7899},[142,179],[91,7901,458],{"className":7902},[142,179],[91,7904,7905,7908],{"style":188},[91,7906],{"className":7907,"style":172},[171],[91,7909],{"className":7910,"style":196},[195],[91,7912,7913,7916],{"style":199},[91,7914],{"className":7915,"style":172},[171],[91,7917,7919],{"className":7918},[176,177,178,179],[91,7920,7922],{"className":7921},[142,179],[91,7923,326],{"className":7924},[142,179],[91,7926,218],{"className":7927},[217],[91,7929,7931],{"className":7930},[159],[91,7932,7934],{"className":7933,"style":225},[163],[91,7935],{},[91,7937],{"className":7938},[231,147],[91,7940],{"className":7941,"style":2275},[420],[91,7943,6524],{"className":7944},[2279],[91,7946],{"className":7947,"style":2275},[420],[91,7949,7951,7954,7957,7960,7963],{"className":7950},[133],[91,7952],{"className":7953,"style":1383},[137],[91,7955,326],{"className":7956},[142],[91,7958],{"className":7959,"style":2275},[420],[91,7961,6524],{"className":7962},[2279],[91,7964],{"className":7965,"style":2275},[420],[91,7967,7969,7972,7975,7978,7981],{"className":7968},[133],[91,7970],{"className":7971,"style":435},[137],[91,7973,326],{"className":7974},[142],[91,7976],{"className":7977,"style":421},[420],[91,7979,332],{"className":7980},[425],[91,7982],{"className":7983,"style":421},[420],[91,7985,7987,7990,7993,7999],{"className":7986},[133],[91,7988],{"className":7989,"style":567},[137],[91,7991,571],{"className":7992},[142],[91,7994,7996],{"className":7995},[142],[91,7997,579],{"className":7998},[578],[91,8000,583],{"className":8001},[142],[2307,8003,8004],{"align":7816},[91,8005,8007,8022],{"className":8006},[94],[91,8008,8010],{"className":8009},[98],[100,8011,8012],{"xmlns":102},[104,8013,8014,8019],{},[107,8015,8016],{},[117,8017,8018],{},"0,5000",[121,8020,8021],{"encoding":123},"0{,}5000",[91,8023,8025],{"className":8024,"ariaHidden":129},[128],[91,8026,8028,8031,8034,8040],{"className":8027},[133],[91,8029],{"className":8030,"style":567},[137],[91,8032,571],{"className":8033},[142],[91,8035,8037],{"className":8036},[142],[91,8038,579],{"className":8039},[578],[91,8041,8043],{"className":8042},[142],"5000",[2226,8045,8046,8048,8295],{},[2307,8047,326],{"align":2231},[2307,8049,8050],{"align":7816},[91,8051,8053,8089],{"className":8052},[94],[91,8054,8056],{"className":8055},[98],[100,8057,8058],{"xmlns":102},[104,8059,8060,8086],{},[107,8061,8062,8068,8070,8072,8074,8080,8083],{},[110,8063,8064,8066],{},[117,8065,326],{},[117,8067,1101],{},[330,8069,6524],{},[117,8071,326],{},[330,8073,6524],{},[110,8075,8076,8078],{},[117,8077,458],{},[117,8079,980],{},[330,8081,8082],{},"≈",[117,8084,8085],{},"0,0833",[121,8087,8088],{"encoding":123},"\\frac{1}{8} \\times 1 \\times \\frac{2}{3} \\approx 0{,}0833",[91,8090,8092,8175,8193,8276],{"className":8091,"ariaHidden":129},[128],[91,8093,8095,8098,8166,8169,8172],{"className":8094},[133],[91,8096],{"className":8097,"style":347},[137],[91,8099,8101,8104,8163],{"className":8100},[142],[91,8102],{"className":8103},[146,147],[91,8105,8107],{"className":8106},[110],[91,8108,8110,8155],{"className":8109},[154,155],[91,8111,8113,8152],{"className":8112},[159],[91,8114,8116,8130,8138],{"className":8115,"style":366},[163],[91,8117,8118,8121],{"style":167},[91,8119],{"className":8120,"style":172},[171],[91,8122,8124],{"className":8123},[176,177,178,179],[91,8125,8127],{"className":8126},[142,179],[91,8128,1101],{"className":8129},[142,179],[91,8131,8132,8135],{"style":188},[91,8133],{"className":8134,"style":172},[171],[91,8136],{"className":8137,"style":196},[195],[91,8139,8140,8143],{"style":199},[91,8141],{"className":8142,"style":172},[171],[91,8144,8146],{"className":8145},[176,177,178,179],[91,8147,8149],{"className":8148},[142,179],[91,8150,326],{"className":8151},[142,179],[91,8153,218],{"className":8154},[217],[91,8156,8158],{"className":8157},[159],[91,8159,8161],{"className":8160,"style":225},[163],[91,8162],{},[91,8164],{"className":8165},[231,147],[91,8167],{"className":8168,"style":2275},[420],[91,8170,6524],{"className":8171},[2279],[91,8173],{"className":8174,"style":2275},[420],[91,8176,8178,8181,8184,8187,8190],{"className":8177},[133],[91,8179],{"className":8180,"style":1383},[137],[91,8182,326],{"className":8183},[142],[91,8185],{"className":8186,"style":2275},[420],[91,8188,6524],{"className":8189},[2279],[91,8191],{"className":8192,"style":2275},[420],[91,8194,8196,8199,8267,8270,8273],{"className":8195},[133],[91,8197],{"className":8198,"style":347},[137],[91,8200,8202,8205,8264],{"className":8201},[142],[91,8203],{"className":8204},[146,147],[91,8206,8208],{"className":8207},[110],[91,8209,8211,8256],{"className":8210},[154,155],[91,8212,8214,8253],{"className":8213},[159],[91,8215,8217,8231,8239],{"className":8216,"style":366},[163],[91,8218,8219,8222],{"style":167},[91,8220],{"className":8221,"style":172},[171],[91,8223,8225],{"className":8224},[176,177,178,179],[91,8226,8228],{"className":8227},[142,179],[91,8229,980],{"className":8230},[142,179],[91,8232,8233,8236],{"style":188},[91,8234],{"className":8235,"style":172},[171],[91,8237],{"className":8238,"style":196},[195],[91,8240,8241,8244],{"style":199},[91,8242],{"className":8243,"style":172},[171],[91,8245,8247],{"className":8246},[176,177,178,179],[91,8248,8250],{"className":8249},[142,179],[91,8251,458],{"className":8252},[142,179],[91,8254,218],{"className":8255},[217],[91,8257,8259],{"className":8258},[159],[91,8260,8262],{"className":8261,"style":225},[163],[91,8263],{},[91,8265],{"className":8266},[231,147],[91,8268],{"className":8269,"style":421},[420],[91,8271,8082],{"className":8272},[425],[91,8274],{"className":8275,"style":421},[420],[91,8277,8279,8282,8285,8291],{"className":8278},[133],[91,8280],{"className":8281,"style":567},[137],[91,8283,571],{"className":8284},[142],[91,8286,8288],{"className":8287},[142],[91,8289,579],{"className":8290},[578],[91,8292,8294],{"className":8293},[142],"0833",[2307,8296,8297],{"align":7816},[91,8298,8300,8315],{"className":8299},[94],[91,8301,8303],{"className":8302},[98],[100,8304,8305],{"xmlns":102},[104,8306,8307,8312],{},[107,8308,8309],{},[117,8310,8311],{},"0,5833",[121,8313,8314],{"encoding":123},"0{,}5833",[91,8316,8318],{"className":8317,"ariaHidden":129},[128],[91,8319,8321,8324,8327,8333],{"className":8320},[133],[91,8322],{"className":8323,"style":567},[137],[91,8325,571],{"className":8326},[142],[91,8328,8330],{"className":8329},[142],[91,8331,579],{"className":8332},[578],[91,8334,8336],{"className":8335},[142],"5833",[2226,8338,8339,8341,8587],{},[2307,8340,458],{"align":2231},[2307,8342,8343],{"align":7816},[91,8344,8346,8381],{"className":8345},[94],[91,8347,8349],{"className":8348},[98],[100,8350,8351],{"xmlns":102},[104,8352,8353,8378],{},[107,8354,8355,8361,8363,8365,8367,8373,8375],{},[110,8356,8357,8359],{},[117,8358,326],{},[117,8360,1899],{},[330,8362,6524],{},[117,8364,458],{},[330,8366,6524],{},[110,8368,8369,8371],{},[117,8370,980],{},[117,8372,583],{},[330,8374,332],{},[117,8376,8377],{},"0,0375",[121,8379,8380],{"encoding":123},"\\frac{1}{32} \\times 2 \\times \\frac{3}{5} = 0{,}0375",[91,8382,8384,8467,8485,8568],{"className":8383,"ariaHidden":129},[128],[91,8385,8387,8390,8458,8461,8464],{"className":8386},[133],[91,8388],{"className":8389,"style":347},[137],[91,8391,8393,8396,8455],{"className":8392},[142],[91,8394],{"className":8395},[146,147],[91,8397,8399],{"className":8398},[110],[91,8400,8402,8447],{"className":8401},[154,155],[91,8403,8405,8444],{"className":8404},[159],[91,8406,8408,8422,8430],{"className":8407,"style":366},[163],[91,8409,8410,8413],{"style":167},[91,8411],{"className":8412,"style":172},[171],[91,8414,8416],{"className":8415},[176,177,178,179],[91,8417,8419],{"className":8418},[142,179],[91,8420,1899],{"className":8421},[142,179],[91,8423,8424,8427],{"style":188},[91,8425],{"className":8426,"style":172},[171],[91,8428],{"className":8429,"style":196},[195],[91,8431,8432,8435],{"style":199},[91,8433],{"className":8434,"style":172},[171],[91,8436,8438],{"className":8437},[176,177,178,179],[91,8439,8441],{"className":8440},[142,179],[91,8442,326],{"className":8443},[142,179],[91,8445,218],{"className":8446},[217],[91,8448,8450],{"className":8449},[159],[91,8451,8453],{"className":8452,"style":225},[163],[91,8454],{},[91,8456],{"className":8457},[231,147],[91,8459],{"className":8460,"style":2275},[420],[91,8462,6524],{"className":8463},[2279],[91,8465],{"className":8466,"style":2275},[420],[91,8468,8470,8473,8476,8479,8482],{"className":8469},[133],[91,8471],{"className":8472,"style":1383},[137],[91,8474,458],{"className":8475},[142],[91,8477],{"className":8478,"style":2275},[420],[91,8480,6524],{"className":8481},[2279],[91,8483],{"className":8484,"style":2275},[420],[91,8486,8488,8491,8559,8562,8565],{"className":8487},[133],[91,8489],{"className":8490,"style":347},[137],[91,8492,8494,8497,8556],{"className":8493},[142],[91,8495],{"className":8496},[146,147],[91,8498,8500],{"className":8499},[110],[91,8501,8503,8548],{"className":8502},[154,155],[91,8504,8506,8545],{"className":8505},[159],[91,8507,8509,8523,8531],{"className":8508,"style":366},[163],[91,8510,8511,8514],{"style":167},[91,8512],{"className":8513,"style":172},[171],[91,8515,8517],{"className":8516},[176,177,178,179],[91,8518,8520],{"className":8519},[142,179],[91,8521,583],{"className":8522},[142,179],[91,8524,8525,8528],{"style":188},[91,8526],{"className":8527,"style":172},[171],[91,8529],{"className":8530,"style":196},[195],[91,8532,8533,8536],{"style":199},[91,8534],{"className":8535,"style":172},[171],[91,8537,8539],{"className":8538},[176,177,178,179],[91,8540,8542],{"className":8541},[142,179],[91,8543,980],{"className":8544},[142,179],[91,8546,218],{"className":8547},[217],[91,8549,8551],{"className":8550},[159],[91,8552,8554],{"className":8553,"style":225},[163],[91,8555],{},[91,8557],{"className":8558},[231,147],[91,8560],{"className":8561,"style":421},[420],[91,8563,332],{"className":8564},[425],[91,8566],{"className":8567,"style":421},[420],[91,8569,8571,8574,8577,8583],{"className":8570},[133],[91,8572],{"className":8573,"style":567},[137],[91,8575,571],{"className":8576},[142],[91,8578,8580],{"className":8579},[142],[91,8581,579],{"className":8582},[578],[91,8584,8586],{"className":8585},[142],"0375",[2307,8588,8589],{"align":7816},[91,8590,8592,8607],{"className":8591},[94],[91,8593,8595],{"className":8594},[98],[100,8596,8597],{"xmlns":102},[104,8598,8599,8604],{},[107,8600,8601],{},[117,8602,8603],{},"0,6208",[121,8605,8606],{"encoding":123},"0{,}6208",[91,8608,8610],{"className":8609,"ariaHidden":129},[128],[91,8611,8613,8616,8619,8625],{"className":8612},[133],[91,8614],{"className":8615,"style":567},[137],[91,8617,571],{"className":8618},[142],[91,8620,8622],{"className":8621},[142],[91,8623,579],{"className":8624},[578],[91,8626,8628],{"className":8627},[142],"6208",[2226,8630,8631,8633,8879],{},[2307,8632,980],{"align":2231},[2307,8634,8635],{"align":7816},[91,8636,8638,8673],{"className":8637},[94],[91,8639,8641],{"className":8640},[98],[100,8642,8643],{"xmlns":102},[104,8644,8645,8670],{},[107,8646,8647,8653,8655,8657,8659,8665,8667],{},[110,8648,8649,8651],{},[117,8650,326],{},[117,8652,3255],{},[330,8654,6524],{},[117,8656,583],{},[330,8658,6524],{},[110,8660,8661,8663],{},[117,8662,119],{},[117,8664,3152],{},[330,8666,8082],{},[117,8668,8669],{},"0,0223",[121,8671,8672],{"encoding":123},"\\frac{1}{128} \\times 5 \\times \\frac{4}{7} \\approx 0{,}0223",[91,8674,8676,8759,8777,8860],{"className":8675,"ariaHidden":129},[128],[91,8677,8679,8682,8750,8753,8756],{"className":8678},[133],[91,8680],{"className":8681,"style":347},[137],[91,8683,8685,8688,8747],{"className":8684},[142],[91,8686],{"className":8687},[146,147],[91,8689,8691],{"className":8690},[110],[91,8692,8694,8739],{"className":8693},[154,155],[91,8695,8697,8736],{"className":8696},[159],[91,8698,8700,8714,8722],{"className":8699,"style":366},[163],[91,8701,8702,8705],{"style":167},[91,8703],{"className":8704,"style":172},[171],[91,8706,8708],{"className":8707},[176,177,178,179],[91,8709,8711],{"className":8710},[142,179],[91,8712,3255],{"className":8713},[142,179],[91,8715,8716,8719],{"style":188},[91,8717],{"className":8718,"style":172},[171],[91,8720],{"className":8721,"style":196},[195],[91,8723,8724,8727],{"style":199},[91,8725],{"className":8726,"style":172},[171],[91,8728,8730],{"className":8729},[176,177,178,179],[91,8731,8733],{"className":8732},[142,179],[91,8734,326],{"className":8735},[142,179],[91,8737,218],{"className":8738},[217],[91,8740,8742],{"className":8741},[159],[91,8743,8745],{"className":8744,"style":225},[163],[91,8746],{},[91,8748],{"className":8749},[231,147],[91,8751],{"className":8752,"style":2275},[420],[91,8754,6524],{"className":8755},[2279],[91,8757],{"className":8758,"style":2275},[420],[91,8760,8762,8765,8768,8771,8774],{"className":8761},[133],[91,8763],{"className":8764,"style":1383},[137],[91,8766,583],{"className":8767},[142],[91,8769],{"className":8770,"style":2275},[420],[91,8772,6524],{"className":8773},[2279],[91,8775],{"className":8776,"style":2275},[420],[91,8778,8780,8783,8851,8854,8857],{"className":8779},[133],[91,8781],{"className":8782,"style":347},[137],[91,8784,8786,8789,8848],{"className":8785},[142],[91,8787],{"className":8788},[146,147],[91,8790,8792],{"className":8791},[110],[91,8793,8795,8840],{"className":8794},[154,155],[91,8796,8798,8837],{"className":8797},[159],[91,8799,8801,8815,8823],{"className":8800,"style":366},[163],[91,8802,8803,8806],{"style":167},[91,8804],{"className":8805,"style":172},[171],[91,8807,8809],{"className":8808},[176,177,178,179],[91,8810,8812],{"className":8811},[142,179],[91,8813,3152],{"className":8814},[142,179],[91,8816,8817,8820],{"style":188},[91,8818],{"className":8819,"style":172},[171],[91,8821],{"className":8822,"style":196},[195],[91,8824,8825,8828],{"style":199},[91,8826],{"className":8827,"style":172},[171],[91,8829,8831],{"className":8830},[176,177,178,179],[91,8832,8834],{"className":8833},[142,179],[91,8835,119],{"className":8836},[142,179],[91,8838,218],{"className":8839},[217],[91,8841,8843],{"className":8842},[159],[91,8844,8846],{"className":8845,"style":225},[163],[91,8847],{},[91,8849],{"className":8850},[231,147],[91,8852],{"className":8853,"style":421},[420],[91,8855,8082],{"className":8856},[425],[91,8858],{"className":8859,"style":421},[420],[91,8861,8863,8866,8869,8875],{"className":8862},[133],[91,8864],{"className":8865,"style":567},[137],[91,8867,571],{"className":8868},[142],[91,8870,8872],{"className":8871},[142],[91,8873,579],{"className":8874},[578],[91,8876,8878],{"className":8877},[142],"0223",[2307,8880,8881],{"align":7816},[91,8882,8884,8899],{"className":8883},[94],[91,8885,8887],{"className":8886},[98],[100,8888,8889],{"xmlns":102},[104,8890,8891,8896],{},[107,8892,8893],{},[117,8894,8895],{},"0,6431",[121,8897,8898],{"encoding":123},"0{,}6431",[91,8900,8902],{"className":8901,"ariaHidden":129},[128],[91,8903,8905,8908,8911,8917],{"className":8904},[133],[91,8906],{"className":8907,"style":567},[137],[91,8909,571],{"className":8910},[142],[91,8912,8914],{"className":8913},[142],[91,8915,579],{"className":8916},[578],[91,8918,8920],{"className":8919},[142],"6431",[2226,8922,8923,8925,9171],{},[2307,8924,119],{"align":2231},[2307,8926,8927],{"align":7816},[91,8928,8930,8965],{"className":8929},[94],[91,8931,8933],{"className":8932},[98],[100,8934,8935],{"xmlns":102},[104,8936,8937,8962],{},[107,8938,8939,8945,8947,8949,8951,8957,8959],{},[110,8940,8941,8943],{},[117,8942,326],{},[117,8944,3516],{},[330,8946,6524],{},[117,8948,3495],{},[330,8950,6524],{},[110,8952,8953,8955],{},[117,8954,583],{},[117,8956,3412],{},[330,8958,8082],{},[117,8960,8961],{},"0,0152",[121,8963,8964],{"encoding":123},"\\frac{1}{512} \\times 14 \\times \\frac{5}{9} \\approx 0{,}0152",[91,8966,8968,9051,9069,9152],{"className":8967,"ariaHidden":129},[128],[91,8969,8971,8974,9042,9045,9048],{"className":8970},[133],[91,8972],{"className":8973,"style":347},[137],[91,8975,8977,8980,9039],{"className":8976},[142],[91,8978],{"className":8979},[146,147],[91,8981,8983],{"className":8982},[110],[91,8984,8986,9031],{"className":8985},[154,155],[91,8987,8989,9028],{"className":8988},[159],[91,8990,8992,9006,9014],{"className":8991,"style":366},[163],[91,8993,8994,8997],{"style":167},[91,8995],{"className":8996,"style":172},[171],[91,8998,9000],{"className":8999},[176,177,178,179],[91,9001,9003],{"className":9002},[142,179],[91,9004,3516],{"className":9005},[142,179],[91,9007,9008,9011],{"style":188},[91,9009],{"className":9010,"style":172},[171],[91,9012],{"className":9013,"style":196},[195],[91,9015,9016,9019],{"style":199},[91,9017],{"className":9018,"style":172},[171],[91,9020,9022],{"className":9021},[176,177,178,179],[91,9023,9025],{"className":9024},[142,179],[91,9026,326],{"className":9027},[142,179],[91,9029,218],{"className":9030},[217],[91,9032,9034],{"className":9033},[159],[91,9035,9037],{"className":9036,"style":225},[163],[91,9038],{},[91,9040],{"className":9041},[231,147],[91,9043],{"className":9044,"style":2275},[420],[91,9046,6524],{"className":9047},[2279],[91,9049],{"className":9050,"style":2275},[420],[91,9052,9054,9057,9060,9063,9066],{"className":9053},[133],[91,9055],{"className":9056,"style":1383},[137],[91,9058,3495],{"className":9059},[142],[91,9061],{"className":9062,"style":2275},[420],[91,9064,6524],{"className":9065},[2279],[91,9067],{"className":9068,"style":2275},[420],[91,9070,9072,9075,9143,9146,9149],{"className":9071},[133],[91,9073],{"className":9074,"style":347},[137],[91,9076,9078,9081,9140],{"className":9077},[142],[91,9079],{"className":9080},[146,147],[91,9082,9084],{"className":9083},[110],[91,9085,9087,9132],{"className":9086},[154,155],[91,9088,9090,9129],{"className":9089},[159],[91,9091,9093,9107,9115],{"className":9092,"style":366},[163],[91,9094,9095,9098],{"style":167},[91,9096],{"className":9097,"style":172},[171],[91,9099,9101],{"className":9100},[176,177,178,179],[91,9102,9104],{"className":9103},[142,179],[91,9105,3412],{"className":9106},[142,179],[91,9108,9109,9112],{"style":188},[91,9110],{"className":9111,"style":172},[171],[91,9113],{"className":9114,"style":196},[195],[91,9116,9117,9120],{"style":199},[91,9118],{"className":9119,"style":172},[171],[91,9121,9123],{"className":9122},[176,177,178,179],[91,9124,9126],{"className":9125},[142,179],[91,9127,583],{"className":9128},[142,179],[91,9130,218],{"className":9131},[217],[91,9133,9135],{"className":9134},[159],[91,9136,9138],{"className":9137,"style":225},[163],[91,9139],{},[91,9141],{"className":9142},[231,147],[91,9144],{"className":9145,"style":421},[420],[91,9147,8082],{"className":9148},[425],[91,9150],{"className":9151,"style":421},[420],[91,9153,9155,9158,9161,9167],{"className":9154},[133],[91,9156],{"className":9157,"style":567},[137],[91,9159,571],{"className":9160},[142],[91,9162,9164],{"className":9163},[142],[91,9165,579],{"className":9166},[578],[91,9168,9170],{"className":9169},[142],"0152",[2307,9172,9173],{"align":7816},[91,9174,9176,9191],{"className":9175},[94],[91,9177,9179],{"className":9178},[98],[100,9180,9181],{"xmlns":102},[104,9182,9183,9188],{},[107,9184,9185],{},[117,9186,9187],{},"0,6583",[121,9189,9190],{"encoding":123},"0{,}6583",[91,9192,9194],{"className":9193,"ariaHidden":129},[128],[91,9195,9197,9200,9203,9209],{"className":9196},[133],[91,9198],{"className":9199,"style":567},[137],[91,9201,571],{"className":9202},[142],[91,9204,9206],{"className":9205},[142],[91,9207,579],{"className":9208},[578],[91,9210,9212],{"className":9211},[142],"6583",[17,9214,9215,9216,9253,9254,9385],{},"Con sólo cinco términos ya estamos en ",[91,9217,9219,9232],{"className":9218},[94],[91,9220,9222],{"className":9221},[98],[100,9223,9224],{"xmlns":102},[104,9225,9226,9230],{},[107,9227,9228],{},[117,9229,9187],{},[121,9231,9190],{"encoding":123},[91,9233,9235],{"className":9234,"ariaHidden":129},[128],[91,9236,9238,9241,9244,9250],{"className":9237},[133],[91,9239],{"className":9240,"style":567},[137],[91,9242,571],{"className":9243},[142],[91,9245,9247],{"className":9246},[142],[91,9248,579],{"className":9249},[578],[91,9251,9212],{"className":9252},[142],", acercándonos lentamente a ",[91,9255,9257,9280],{"className":9256},[94],[91,9258,9260],{"className":9259},[98],[100,9261,9262],{"xmlns":102},[104,9263,9264,9277],{},[107,9265,9266,9272,9274],{},[110,9267,9268,9270],{},[113,9269,115],{},[117,9271,119],{},[330,9273,8082],{},[117,9275,9276],{},"0,7854",[121,9278,9279],{"encoding":123},"\\frac{\\pi}{4} \\approx 0{,}7854",[91,9281,9283,9366],{"className":9282,"ariaHidden":129},[128],[91,9284,9286,9289,9357,9360,9363],{"className":9285},[133],[91,9287],{"className":9288,"style":138},[137],[91,9290,9292,9295,9354],{"className":9291},[142],[91,9293],{"className":9294},[146,147],[91,9296,9298],{"className":9297},[110],[91,9299,9301,9346],{"className":9300},[154,155],[91,9302,9304,9343],{"className":9303},[159],[91,9305,9307,9321,9329],{"className":9306,"style":164},[163],[91,9308,9309,9312],{"style":167},[91,9310],{"className":9311,"style":172},[171],[91,9313,9315],{"className":9314},[176,177,178,179],[91,9316,9318],{"className":9317},[142,179],[91,9319,119],{"className":9320},[142,179],[91,9322,9323,9326],{"style":188},[91,9324],{"className":9325,"style":172},[171],[91,9327],{"className":9328,"style":196},[195],[91,9330,9331,9334],{"style":199},[91,9332],{"className":9333,"style":172},[171],[91,9335,9337],{"className":9336},[176,177,178,179],[91,9338,9340],{"className":9339},[142,179],[91,9341,115],{"className":9342,"style":213},[142,212,179],[91,9344,218],{"className":9345},[217],[91,9347,9349],{"className":9348},[159],[91,9350,9352],{"className":9351,"style":225},[163],[91,9353],{},[91,9355],{"className":9356},[231,147],[91,9358],{"className":9359,"style":421},[420],[91,9361,8082],{"className":9362},[425],[91,9364],{"className":9365,"style":421},[420],[91,9367,9369,9372,9375,9381],{"className":9368},[133],[91,9370],{"className":9371,"style":567},[137],[91,9373,571],{"className":9374},[142],[91,9376,9378],{"className":9377},[142],[91,9379,579],{"className":9380},[578],[91,9382,9384],{"className":9383},[142],"7854",". La convergencia es lenta, pero la dirección es inequívoca cuando el número de lanzamientos es suficientemente grande.",[12,9387,9389],{"id":9388},"la-aparición-del-arcoseno","La aparición del arcoseno",[17,9391,9392],{},"Aquí es donde ocurre la magia. Sustituyamos la fórmula de los números de Catalan en nuestra serie:",[17,9394,9396],{"className":9395},[645],[91,9397,9399,9446],{"className":9398},[94],[91,9400,9402],{"className":9401},[98],[100,9403,9404],{"xmlns":102},[104,9405,9406,9444],{},[107,9407,9408,9414,9416,9428],{},[4346,9409,9410,9412],{},[113,9411,4350],{},[113,9413,2249],{},[330,9415,332],{},[110,9417,9418,9420],{},[117,9419,326],{},[107,9421,9422,9424,9426],{},[113,9423,2249],{},[330,9425,1368],{},[117,9427,326],{},[107,9429,9430,9432,9442],{},[330,9431,1081],{"fence":129},[110,9433,9434,9440],{"linethickness":4373},[107,9435,9436,9438],{},[117,9437,458],{},[113,9439,2249],{},[113,9441,2249],{},[330,9443,1090],{"fence":129},[121,9445,4386],{"encoding":123},[91,9447,9449,9504],{"className":9448,"ariaHidden":129},[128],[91,9450,9452,9455,9495,9498,9501],{"className":9451},[133],[91,9453],{"className":9454,"style":4396},[137],[91,9456,9458,9461],{"className":9457},[142],[91,9459,4350],{"className":9460,"style":4403},[142,212],[91,9462,9464],{"className":9463},[1209],[91,9465,9467,9487],{"className":9466},[154,155],[91,9468,9470,9484],{"className":9469},[159],[91,9471,9473],{"className":9472,"style":4416},[163],[91,9474,9475,9478],{"style":4419},[91,9476],{"className":9477,"style":1226},[171],[91,9479,9481],{"className":9480},[176,177,178,179],[91,9482,2249],{"className":9483},[142,212,179],[91,9485,218],{"className":9486},[217],[91,9488,9490],{"className":9489},[159],[91,9491,9493],{"className":9492,"style":4438},[163],[91,9494],{},[91,9496],{"className":9497,"style":421},[420],[91,9499,332],{"className":9500},[425],[91,9502],{"className":9503,"style":421},[420],[91,9505,9507,9510,9584],{"className":9506},[133],[91,9508],{"className":9509,"style":4456},[137],[91,9511,9513,9516,9581],{"className":9512},[142],[91,9514],{"className":9515},[146,147],[91,9517,9519],{"className":9518},[110],[91,9520,9522,9573],{"className":9521},[154,155],[91,9523,9525,9570],{"className":9524},[159],[91,9526,9528,9548,9556],{"className":9527,"style":366},[163],[91,9529,9530,9533],{"style":167},[91,9531],{"className":9532,"style":172},[171],[91,9534,9536],{"className":9535},[176,177,178,179],[91,9537,9539,9542,9545],{"className":9538},[142,179],[91,9540,2249],{"className":9541},[142,212,179],[91,9543,1368],{"className":9544},[2279,179],[91,9546,326],{"className":9547},[142,179],[91,9549,9550,9553],{"style":188},[91,9551],{"className":9552,"style":172},[171],[91,9554],{"className":9555,"style":196},[195],[91,9557,9558,9561],{"style":199},[91,9559],{"className":9560,"style":172},[171],[91,9562,9564],{"className":9563},[176,177,178,179],[91,9565,9567],{"className":9566},[142,179],[91,9568,326],{"className":9569},[142,179],[91,9571,218],{"className":9572},[217],[91,9574,9576],{"className":9575},[159],[91,9577,9579],{"className":9578,"style":3788},[163],[91,9580],{},[91,9582],{"className":9583},[231,147],[91,9585,9587,9593,9647],{"className":9586},[142],[91,9588,9590],{"className":9589,"style":1126},[146,1125],[91,9591,1081],{"className":9592},[1130,1131],[91,9594,9596],{"className":9595},[110],[91,9597,9599,9639],{"className":9598},[154,155],[91,9600,9602,9636],{"className":9601},[159],[91,9603,9605,9619],{"className":9604,"style":4552},[163],[91,9606,9607,9610],{"style":4555},[91,9608],{"className":9609,"style":1226},[171],[91,9611,9613],{"className":9612},[176,177,178,179],[91,9614,9616],{"className":9615},[142,179],[91,9617,2249],{"className":9618},[142,212,179],[91,9620,9621,9624],{"style":4570},[91,9622],{"className":9623,"style":1226},[171],[91,9625,9627],{"className":9626},[176,177,178,179],[91,9628,9630,9633],{"className":9629},[142,179],[91,9631,458],{"className":9632},[142,179],[91,9634,2249],{"className":9635},[142,212,179],[91,9637,218],{"className":9638},[217],[91,9640,9642],{"className":9641},[159],[91,9643,9645],{"className":9644,"style":225},[163],[91,9646],{},[91,9648,9650],{"className":9649,"style":1126},[231,1125],[91,9651,1090],{"className":9652},[1130,1131],[17,9654,9655,9656,9726,9727,9853,9854,9917],{},"Al multiplicar 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por 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el factor ",[91,9855,9857,9880],{"className":9856},[94],[91,9858,9860],{"className":9859},[98],[100,9861,9862],{"xmlns":102},[104,9863,9864,9877],{},[107,9865,9866,9869,9871,9873,9875],{},[330,9867,1081],{"stretchy":9868},"false",[113,9870,2249],{},[330,9872,1368],{},[117,9874,326],{},[330,9876,1090],{"stretchy":9868},[121,9878,9879],{"encoding":123},"(n+1)",[91,9881,9883,9905],{"className":9882,"ariaHidden":129},[128],[91,9884,9886,9890,9893,9896,9899,9902],{"className":9885},[133],[91,9887],{"className":9888,"style":9889},[137],"height:1em;vertical-align:-0.25em;",[91,9891,1081],{"className":9892},[146],[91,9894,2249],{"className":9895},[142,212],[91,9897],{"className":9898,"style":2275},[420],[91,9900,1368],{"className":9901},[2279],[91,9903],{"className":9904,"style":2275},[420],[91,9906,9908,9911,9914],{"className":9907},[133],[91,9909],{"className":9910,"style":9889},[137],[91,9912,326],{"className":9913},[142],[91,9915,1090],{"className":9916},[231]," se cancela limpiamente:",[17,9919,9921],{"className":9920},[645],[91,9922,9924,10016],{"className":9923},[94],[91,9925,9927],{"className":9926},[98],[100,9928,9929],{"xmlns":102},[104,9930,9931,10013],{},[107,9932,9933,9945,9961,9963,9983,9985],{},[110,9934,9935,9937],{},[117,9936,326],{},[107,9938,9939,9941,9943],{},[113,9940,2249],{},[330,9942,1368],{},[117,9944,326],{},[107,9946,9947,9949,9959],{},[330,9948,1081],{"fence":129},[110,9950,9951,9957],{"linethickness":4373},[107,9952,9953,9955],{},[117,9954,458],{},[113,9956,2249],{},[113,9958,2249],{},[330,9960,1090],{"fence":129},[330,9962,7396],{},[110,9964,9965,9973],{},[107,9966,9967,9969,9971],{},[113,9968,2249],{},[330,9970,1368],{},[117,9972,326],{},[107,9974,9975,9977,9979,9981],{},[117,9976,458],{},[113,9978,2249],{},[330,9980,1368],{},[117,9982,326],{},[330,9984,332],{},[110,9986,9987,10003],{},[107,9988,9989,9991,10001],{},[330,9990,1081],{"fence":129},[110,9992,9993,9999],{"linethickness":4373},[107,9994,9995,9997],{},[117,9996,458],{},[113,9998,2249],{},[113,10000,2249],{},[330,10002,1090],{"fence":129},[107,10004,10005,10007,10009,10011],{},[117,10006,458],{},[113,10008,2249],{},[330,10010,1368],{},[117,10012,326],{},[121,10014,10015],{"encoding":123},"\\frac{1}{n+1}\\binom{2n}{n} \\cdot \\frac{n+1}{2n+1} = \\frac{\\binom{2n}{n}}{2n+1}",[91,10017,10019,10177,10275],{"className":10018,"ariaHidden":129},[128],[91,10020,10022,10025,10099,10168,10171,10174],{"className":10021},[133],[91,10023],{"className":10024,"style":4456},[137],[91,10026,10028,10031,10096],{"className":10027},[142],[91,10029],{"className":10030},[146,147],[91,10032,10034],{"className":10033},[110],[91,10035,10037,10088],{"className":10036},[154,155],[91,10038,10040,10085],{"className":10039},[159],[91,10041,10043,10063,10071],{"className":10042,"style":366},[163],[91,10044,10045,10048],{"style":167},[91,10046],{"className":10047,"style":172},[171],[91,10049,10051],{"className":10050},[176,177,178,179],[91,10052,10054,10057,10060],{"className":10053},[142,179],[91,10055,2249],{"className":10056},[142,212,179],[91,10058,1368],{"className":10059},[2279,179],[91,10061,326],{"className":10062},[142,179],[91,10064,10065,10068],{"style":188},[91,10066],{"className":10067,"style":172},[171],[91,10069],{"className":10070,"style":196},[195],[91,10072,10073,10076],{"style":199},[91,10074],{"className":10075,"style":172},[171],[91,10077,10079],{"className":10078},[176,177,178,179],[91,10080,10082],{"className":10081},[142,179],[91,10083,326],{"className":10084},[142,179],[91,10086,218],{"className":10087},[217],[91,10089,10091],{"className":10090},[159],[91,10092,10094],{"className":10093,"style":3788},[163],[91,10095],{},[91,10097],{"className":10098},[231,147],[91,10100,10102,10108,10162],{"className":10101},[142],[91,10103,10105],{"className":10104,"style":1126},[146,1125],[91,10106,1081],{"className":10107},[1130,1131],[91,10109,10111],{"className":10110},[110],[91,10112,10114,10154],{"className":10113},[154,155],[91,10115,10117,10151],{"className":10116},[159],[91,10118,10120,10134],{"className":10119,"style":4552},[163],[91,10121,10122,10125],{"style":4555},[91,10123],{"className":10124,"style":1226},[171],[91,10126,10128],{"className":10127},[176,177,178,179],[91,10129,10131],{"className":10130},[142,179],[91,10132,2249],{"className":10133},[142,212,179],[91,10135,10136,10139],{"style":4570},[91,10137],{"className":10138,"style":1226},[171],[91,10140,10142],{"className":10141},[176,177,178,179],[91,10143,10145,10148],{"className":10144},[142,179],[91,10146,458],{"className":10147},[142,179],[91,10149,2249],{"className":10150},[142,212,179],[91,10152,218],{"className":10153},[217],[91,10155,10157],{"className":10156},[159],[91,10158,10160],{"className":10159,"style":225},[163],[91,10161],{},[91,10163,10165],{"className":10164,"style":1126},[231,1125],[91,10166,1090],{"className":10167},[1130,1131],[91,10169],{"className":10170,"style":2275},[420],[91,10172,7396],{"className":10173},[2279],[91,10175],{"className":10176,"style":2275},[420],[91,10178,10180,10183,10266,10269,10272],{"className":10179},[133],[91,10181],{"className":10182,"style":3709},[137],[91,10184,10186,10189,10263],{"className":10185},[142],[91,10187],{"className":10188},[146,147],[91,10190,10192],{"className":10191},[110],[91,10193,10195,10255],{"className":10194},[154,155],[91,10196,10198,10252],{"className":10197},[159],[91,10199,10201,10224,10232],{"className":10200,"style":366},[163],[91,10202,10203,10206],{"style":167},[91,10204],{"className":10205,"style":172},[171],[91,10207,10209],{"className":10208},[176,177,178,179],[91,10210,10212,10215,10218,10221],{"className":10211},[142,179],[91,10213,458],{"className":10214},[142,179],[91,10216,2249],{"className":10217},[142,212,179],[91,10219,1368],{"className":10220},[2279,179],[91,10222,326],{"className":10223},[142,179],[91,10225,10226,10229],{"style":188},[91,10227],{"className":10228,"style":172},[171],[91,10230],{"className":10231,"style":196},[195],[91,10233,10234,10237],{"style":199},[91,10235],{"className":10236,"style":172},[171],[91,10238,10240],{"className":10239},[176,177,178,179],[91,10241,10243,10246,10249],{"className":10242},[142,179],[91,10244,2249],{"className":10245},[142,212,179],[91,10247,1368],{"className":10248},[2279,179],[91,10250,326],{"className":10251},[142,179],[91,10253,218],{"className":10254},[217],[91,10256,10258],{"className":10257},[159],[91,10259,10261],{"className":10260,"style":3788},[163],[91,10262],{},[91,10264],{"className":10265},[231,147],[91,10267],{"className":10268,"style":421},[420],[91,10270,332],{"className":10271},[425],[91,10273],{"className":10274,"style":421},[420],[91,10276,10278,10282],{"className":10277},[133],[91,10279],{"className":10280,"style":10281},[137],"height:1.9133em;vertical-align:-0.4033em;",[91,10283,10285,10288,10436],{"className":10284},[142],[91,10286],{"className":10287},[146,147],[91,10289,10291],{"className":10290},[110],[91,10292,10294,10428],{"className":10293},[154,155],[91,10295,10297,10425],{"className":10296},[159],[91,10298,10301,10324,10332],{"className":10299,"style":10300},[163],"height:1.51em;",[91,10302,10303,10306],{"style":167},[91,10304],{"className":10305,"style":172},[171],[91,10307,10309],{"className":10308},[176,177,178,179],[91,10310,10312,10315,10318,10321],{"className":10311},[142,179],[91,10313,458],{"className":10314},[142,179],[91,10316,2249],{"className":10317},[142,212,179],[91,10319,1368],{"className":10320},[2279,179],[91,10322,326],{"className":10323},[142,179],[91,10325,10326,10329],{"style":188},[91,10327],{"className":10328,"style":172},[171],[91,10330],{"className":10331,"style":196},[195],[91,10333,10335,10338],{"style":10334},"top:-3.7125em;",[91,10336],{"className":10337,"style":172},[171],[91,10339,10341],{"className":10340},[176,177,178,179],[91,10342,10344],{"className":10343},[142,179],[91,10345,10347,10358,10416],{"className":10346},[142,179],[91,10348,10352],{"className":10349,"style":10351},[146,176,7584,10350,179,1125],"size6","top:0.075em;",[91,10353,10355],{"className":10354},[1130,1131,179],[91,10356,1081],{"className":10357},[179],[91,10359,10361],{"className":10360},[110],[91,10362,10364,10407],{"className":10363},[154,155],[91,10365,10367,10404],{"className":10366},[159],[91,10368,10371,10386],{"className":10369,"style":10370},[163],"height:0.9313em;",[91,10372,10374,10377],{"style":10373},"top:-2.156em;",[91,10375],{"className":10376,"style":7580},[171],[91,10378,10380],{"className":10379},[176,7584,1131,179],[91,10381,10383],{"className":10382},[142,179],[91,10384,2249],{"className":10385},[142,212,179],[91,10387,10389,10392],{"style":10388},"top:-2.971em;",[91,10390],{"className":10391,"style":7580},[171],[91,10393,10395],{"className":10394},[176,7584,1131,179],[91,10396,10398,10401],{"className":10397},[142,179],[91,10399,458],{"className":10400},[142,179],[91,10402,2249],{"className":10403},[142,212,179],[91,10405,218],{"className":10406},[217],[91,10408,10410],{"className":10409},[159],[91,10411,10414],{"className":10412,"style":10413},[163],"height:0.344em;",[91,10415],{},[91,10417,10419],{"className":10418,"style":10351},[231,176,7584,10350,179,1125],[91,10420,10422],{"className":10421},[1130,1131,179],[91,10423,1090],{"className":10424},[179],[91,10426,218],{"className":10427},[217],[91,10429,10431],{"className":10430},[159],[91,10432,10434],{"className":10433,"style":3788},[163],[91,10435],{},[91,10437],{"className":10438},[231,147],[17,10440,10441],{},"Y la serie queda:",[17,10443,10445],{"className":10444},[645],[91,10446,10448,10584],{"className":10447},[94],[91,10449,10451],{"className":10450},[98],[100,10452,10453],{"xmlns":102},[104,10454,10455,10581],{},[107,10456,10457,10459,10461,10475,10521,10523,10529,10543],{},[659,10458,7351],{},[330,10460,332],{},[7355,10462,10463,10465,10473],{},[330,10464,7359],{},[107,10466,10467,10469,10471],{},[113,10468,2249],{},[330,10470,332],{},[117,10472,571],{},[113,10474,7371],{"mathvariant":7370},[110,10476,10477,10493],{},[107,10478,10479,10481,10491],{},[330,10480,1081],{"fence":129},[110,10482,10483,10489],{"linethickness":4373},[107,10484,10485,10487],{},[117,10486,458],{},[113,10488,2249],{},[113,10490,2249],{},[330,10492,1090],{"fence":129},[107,10494,10495,10509,10511,10513,10515,10517,10519],{},[1075,10496,10497,10499],{},[117,10498,458],{},[107,10500,10501,10503,10505,10507],{},[117,10502,458],{},[113,10504,2249],{},[330,10506,1368],{},[117,10508,326],{},[330,10510,1081],{"stretchy":9868},[117,10512,458],{},[113,10514,2249],{},[330,10516,1368],{},[117,10518,326],{},[330,10520,1090],{"stretchy":9868},[330,10522,332],{},[110,10524,10525,10527],{},[117,10526,326],{},[117,10528,458],{},[7355,10530,10531,10533,10541],{},[330,10532,7359],{},[107,10534,10535,10537,10539],{},[113,10536,2249],{},[330,10538,332],{},[117,10540,571],{},[113,10542,7371],{"mathvariant":7370},[110,10544,10545,10561],{},[107,10546,10547,10549,10559],{},[330,10548,1081],{"fence":129},[110,10550,10551,10557],{"linethickness":4373},[107,10552,10553,10555],{},[117,10554,458],{},[113,10556,2249],{},[113,10558,2249],{},[330,10560,1090],{"fence":129},[107,10562,10563,10569,10571,10573,10575,10577,10579],{},[1075,10564,10565,10567],{},[117,10566,119],{},[113,10568,2249],{},[330,10570,1081],{"stretchy":9868},[117,10572,458],{},[113,10574,2249],{},[330,10576,1368],{},[117,10578,326],{},[330,10580,1090],{"stretchy":9868},[121,10582,10583],{"encoding":123},"\\text{Media} = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{2^{2n+1}(2n+1)} = \\frac{1}{2} \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n(2n+1)}",[91,10585,10587,10608,10887],{"className":10586,"ariaHidden":129},[128],[91,10588,10590,10593,10599,10602,10605],{"className":10589},[133],[91,10591],{"className":10592,"style":7078},[137],[91,10594,10596],{"className":10595},[142,678],[91,10597,7351],{"className":10598},[142],[91,10600],{"className":10601,"style":421},[420],[91,10603,332],{"className":10604},[425],[91,10606],{"className":10607,"style":421},[420],[91,10609,10611,10615,10678,10681,10878,10881,10884],{"className":10610},[133],[91,10612],{"className":10613,"style":10614},[137],"height:2.03em;vertical-align:-0.52em;",[91,10616,10618,10621],{"className":10617},[7454],[91,10619,7359],{"className":10620,"style":7460},[7454,7458,7459],[91,10622,10624],{"className":10623},[1209],[91,10625,10627,10670],{"className":10626},[154,155],[91,10628,10630,10667],{"className":10629},[159],[91,10631,10633,10653],{"className":10632,"style":7473},[163],[91,10634,10635,10638],{"style":7476},[91,10636],{"className":10637,"style":1226},[171],[91,10639,10641],{"className":10640},[176,177,178,179],[91,10642,10644,10647,10650],{"className":10643},[142,179],[91,10645,2249],{"className":10646},[142,212,179],[91,10648,332],{"className":10649},[425,179],[91,10651,571],{"className":10652},[142,179],[91,10654,10655,10658],{"style":7497},[91,10656],{"className":10657,"style":1226},[171],[91,10659,10661],{"className":10660},[176,177,178,179],[91,10662,10664],{"className":10663},[142,179],[91,10665,7371],{"className":10666},[142,179],[91,10668,218],{"className":10669},[217],[91,10671,10673],{"className":10672},[159],[91,10674,10676],{"className":10675,"style":7519},[163],[91,10677],{},[91,10679],{"className":10680,"style":1504},[420],[91,10682,10684,10687,10875],{"className":10683},[142],[91,10685],{"className":10686},[146,147],[91,10688,10690],{"className":10689},[110],[91,10691,10693,10866],{"className":10692},[154,155],[91,10694,10696,10863],{"className":10695},[159],[91,10697,10699,10769,10777],{"className":10698,"style":10300},[163],[91,10700,10701,10704],{"style":167},[91,10702],{"className":10703,"style":172},[171],[91,10705,10707],{"className":10706},[176,177,178,179],[91,10708,10710,10751,10754,10757,10760,10763,10766],{"className":10709},[142,179],[91,10711,10713,10716],{"className":10712},[142,179],[91,10714,458],{"className":10715},[142,179],[91,10717,10719],{"className":10718},[1209],[91,10720,10722],{"className":10721},[154],[91,10723,10725],{"className":10724},[159],[91,10726,10728],{"className":10727,"style":7573},[163],[91,10729,10730,10733],{"style":7576},[91,10731],{"className":10732,"style":7580},[171],[91,10734,10736],{"className":10735},[176,7584,1131,179],[91,10737,10739,10742,10745,10748],{"className":10738},[142,179],[91,10740,458],{"className":10741},[142,179],[91,10743,2249],{"className":10744},[142,212,179],[91,10746,1368],{"className":10747},[2279,179],[91,10749,326],{"className":10750},[142,179],[91,10752,1081],{"className":10753},[146,179],[91,10755,458],{"className":10756},[142,179],[91,10758,2249],{"className":10759},[142,212,179],[91,10761,1368],{"className":10762},[2279,179],[91,10764,326],{"className":10765},[142,179],[91,10767,1090],{"className":10768},[231,179],[91,10770,10771,10774],{"style":188},[91,10772],{"className":10773,"style":172},[171],[91,10775],{"className":10776,"style":196},[195],[91,10778,10779,10782],{"style":10334},[91,10780],{"className":10781,"style":172},[171],[91,10783,10785],{"className":10784},[176,177,178,179],[91,10786,10788],{"className":10787},[142,179],[91,10789,10791,10800,10854],{"className":10790},[142,179],[91,10792,10794],{"className":10793,"style":10351},[146,176,7584,10350,179,1125],[91,10795,10797],{"className":10796},[1130,1131,179],[91,10798,1081],{"className":10799},[179],[91,10801,10803],{"className":10802},[110],[91,10804,10806,10846],{"className":10805},[154,155],[91,10807,10809,10843],{"className":10808},[159],[91,10810,10812,10826],{"className":10811,"style":10370},[163],[91,10813,10814,10817],{"style":10373},[91,10815],{"className":10816,"style":7580},[171],[91,10818,10820],{"className":10819},[176,7584,1131,179],[91,10821,10823],{"className":10822},[142,179],[91,10824,2249],{"className":10825},[142,212,179],[91,10827,10828,10831],{"style":10388},[91,10829],{"className":10830,"style":7580},[171],[91,10832,10834],{"className":10833},[176,7584,1131,179],[91,10835,10837,10840],{"className":10836},[142,179],[91,10838,458],{"className":10839},[142,179],[91,10841,2249],{"className":10842},[142,212,179],[91,10844,218],{"className":10845},[217],[91,10847,10849],{"className":10848},[159],[91,10850,10852],{"className":10851,"style":10413},[163],[91,10853],{},[91,10855,10857],{"className":10856,"style":10351},[231,176,7584,10350,179,1125],[91,10858,10860],{"className":10859},[1130,1131,179],[91,10861,1090],{"className":10862},[179],[91,10864,218],{"className":10865},[217],[91,10867,10869],{"className":10868},[159],[91,10870,10873],{"className":10871,"style":10872},[163],"height:0.52em;",[91,10874],{},[91,10876],{"className":10877},[231,147],[91,10879],{"className":10880,"style":421},[420],[91,10882,332],{"className":10883},[425],[91,10885],{"className":10886,"style":421},[420],[91,10888,10890,10893,10961,10964,11027,11030],{"className":10889},[133],[91,10891],{"className":10892,"style":10614},[137],[91,10894,10896,10899,10958],{"className":10895},[142],[91,10897],{"className":10898},[146,147],[91,10900,10902],{"className":10901},[110],[91,10903,10905,10950],{"className":10904},[154,155],[91,10906,10908,10947],{"className":10907},[159],[91,10909,10911,10925,10933],{"className":10910,"style":366},[163],[91,10912,10913,10916],{"style":167},[91,10914],{"className":10915,"style":172},[171],[91,10917,10919],{"className":10918},[176,177,178,179],[91,10920,10922],{"className":10921},[142,179],[91,10923,458],{"className":10924},[142,179],[91,10926,10927,10930],{"style":188},[91,10928],{"className":10929,"style":172},[171],[91,10931],{"className":10932,"style":196},[195],[91,10934,10935,10938],{"style":199},[91,10936],{"className":10937,"style":172},[171],[91,10939,10941],{"className":10940},[176,177,178,179],[91,10942,10944],{"className":10943},[142,179],[91,10945,326],{"className":10946},[142,179],[91,10948,218],{"className":10949},[217],[91,10951,10953],{"className":10952},[159],[91,10954,10956],{"className":10955,"style":225},[163],[91,10957],{},[91,10959],{"className":10960},[231,147],[91,10962],{"className":10963,"style":1504},[420],[91,10965,10967,10970],{"className":10966},[7454],[91,10968,7359],{"className":10969,"style":7460},[7454,7458,7459],[91,10971,10973],{"className":10972},[1209],[91,10974,10976,11019],{"className":10975},[154,155],[91,10977,10979,11016],{"className":10978},[159],[91,10980,10982,11002],{"className":10981,"style":7473},[163],[91,10983,10984,10987],{"style":7476},[91,10985],{"className":10986,"style":1226},[171],[91,10988,10990],{"className":10989},[176,177,178,179],[91,10991,10993,10996,10999],{"className":10992},[142,179],[91,10994,2249],{"className":10995},[142,212,179],[91,10997,332],{"className":10998},[425,179],[91,11000,571],{"className":11001},[142,179],[91,11003,11004,11007],{"style":7497},[91,11005],{"className":11006,"style":1226},[171],[91,11008,11010],{"className":11009},[176,177,178,179],[91,11011,11013],{"className":11012},[142,179],[91,11014,7371],{"className":11015},[142,179],[91,11017,218],{"className":11018},[217],[91,11020,11022],{"className":11021},[159],[91,11023,11025],{"className":11024,"style":7519},[163],[91,11026],{},[91,11028],{"className":11029,"style":1504},[420],[91,11031,11033,11036,11212],{"className":11032},[142],[91,11034],{"className":11035},[146,147],[91,11037,11039],{"className":11038},[110],[91,11040,11042,11204],{"className":11041},[154,155],[91,11043,11045,11201],{"className":11044},[159],[91,11046,11048,11107,11115],{"className":11047,"style":10300},[163],[91,11049,11050,11053],{"style":167},[91,11051],{"className":11052,"style":172},[171],[91,11054,11056],{"className":11055},[176,177,178,179],[91,11057,11059,11089,11092,11095,11098,11101,11104],{"className":11058},[142,179],[91,11060,11062,11065],{"className":11061},[142,179],[91,11063,119],{"className":11064},[142,179],[91,11066,11068],{"className":11067},[1209],[91,11069,11071],{"className":11070},[154],[91,11072,11074],{"className":11073},[159],[91,11075,11078],{"className":11076,"style":11077},[163],"height:0.5935em;",[91,11079,11080,11083],{"style":7576},[91,11081],{"className":11082,"style":7580},[171],[91,11084,11086],{"className":11085},[176,7584,1131,179],[91,11087,2249],{"className":11088},[142,212,179],[91,11090,1081],{"className":11091},[146,179],[91,11093,458],{"className":11094},[142,179],[91,11096,2249],{"className":11097},[142,212,179],[91,11099,1368],{"className":11100},[2279,179],[91,11102,326],{"className":11103},[142,179],[91,11105,1090],{"className":11106},[231,179],[91,11108,11109,11112],{"style":188},[91,11110],{"className":11111,"style":172},[171],[91,11113],{"className":11114,"style":196},[195],[91,11116,11117,11120],{"style":10334},[91,11118],{"className":11119,"style":172},[171],[91,11121,11123],{"className":11122},[176,177,178,179],[91,11124,11126],{"className":11125},[142,179],[91,11127,11129,11138,11192],{"className":11128},[142,179],[91,11130,11132],{"className":11131,"style":10351},[146,176,7584,10350,179,1125],[91,11133,11135],{"className":11134},[1130,1131,179],[91,11136,1081],{"className":11137},[179],[91,11139,11141],{"className":11140},[110],[91,11142,11144,11184],{"className":11143},[154,155],[91,11145,11147,11181],{"className":11146},[159],[91,11148,11150,11164],{"className":11149,"style":10370},[163],[91,11151,11152,11155],{"style":10373},[91,11153],{"className":11154,"style":7580},[171],[91,11156,11158],{"className":11157},[176,7584,1131,179],[91,11159,11161],{"className":11160},[142,179],[91,11162,2249],{"className":11163},[142,212,179],[91,11165,11166,11169],{"style":10388},[91,11167],{"className":11168,"style":7580},[171],[91,11170,11172],{"className":11171},[176,7584,1131,179],[91,11173,11175,11178],{"className":11174},[142,179],[91,11176,458],{"className":11177},[142,179],[91,11179,2249],{"className":11180},[142,212,179],[91,11182,218],{"className":11183},[217],[91,11185,11187],{"className":11186},[159],[91,11188,11190],{"className":11189,"style":10413},[163],[91,11191],{},[91,11193,11195],{"className":11194,"style":10351},[231,176,7584,10350,179,1125],[91,11196,11198],{"className":11197},[1130,1131,179],[91,11199,1090],{"className":11200},[179],[91,11202,218],{"className":11203},[217],[91,11205,11207],{"className":11206},[159],[91,11208,11210],{"className":11209,"style":10872},[163],[91,11211],{},[91,11213],{"className":11214},[231,147],[17,11216,11217,11218,11221],{},"Ahora viene el momento que a Matt Parker le hace «enfadar de pura incredulidad». Resulta que existe una función clásica del análisis matemático llamada ",[33,11219,11220],{},"arcoseno"," — la función inversa del seno — cuya expansión en serie de potencias es:",[17,11223,11225],{"className":11224},[645],[91,11226,11228,11321],{"className":11227},[94],[91,11229,11231],{"className":11230},[98],[100,11232,11233],{"xmlns":102},[104,11234,11235,11318],{},[107,11236,11237,11240,11243,11245,11248,11250,11252,11266,11290,11292],{},[113,11238,11239],{},"arcsin",[330,11241,11242],{},"⁡",[330,11244,1081],{"stretchy":9868},[113,11246,11247],{},"x",[330,11249,1090],{"stretchy":9868},[330,11251,332],{},[7355,11253,11254,11256,11264],{},[330,11255,7359],{},[107,11257,11258,11260,11262],{},[113,11259,2249],{},[330,11261,332],{},[117,11263,571],{},[113,11265,7371],{"mathvariant":7370},[110,11267,11268,11284],{},[107,11269,11270,11272,11282],{},[330,11271,1081],{"fence":129},[110,11273,11274,11280],{"linethickness":4373},[107,11275,11276,11278],{},[117,11277,458],{},[113,11279,2249],{},[113,11281,2249],{},[330,11283,1090],{"fence":129},[1075,11285,11286,11288],{},[117,11287,119],{},[113,11289,2249],{},[330,11291,7396],{},[110,11293,11294,11308],{},[1075,11295,11296,11298],{},[113,11297,11247],{},[107,11299,11300,11302,11304,11306],{},[117,11301,458],{},[113,11303,2249],{},[330,11305,1368],{},[117,11307,326],{},[107,11309,11310,11312,11314,11316],{},[117,11311,458],{},[113,11313,2249],{},[330,11315,1368],{},[117,11317,326],{},[121,11319,11320],{"encoding":123},"\\arcsin(x) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n} \\cdot \\frac{x^{2n+1}}{2n+1}",[91,11322,11324,11351,11599],{"className":11323,"ariaHidden":129},[128],[91,11325,11327,11330,11333,11336,11339,11342,11345,11348],{"className":11326},[133],[91,11328],{"className":11329,"style":9889},[137],[91,11331,11239],{"className":11332},[7454],[91,11334,1081],{"className":11335},[146],[91,11337,11247],{"className":11338},[142,212],[91,11340,1090],{"className":11341},[231],[91,11343],{"className":11344,"style":421},[420],[91,11346,332],{"className":11347},[425],[91,11349],{"className":11350,"style":421},[420],[91,11352,11354,11358,11421,11424,11590,11593,11596],{"className":11353},[133],[91,11355],{"className":11356,"style":11357},[137],"height:1.855em;vertical-align:-0.345em;",[91,11359,11361,11364],{"className":11360},[7454],[91,11362,7359],{"className":11363,"style":7460},[7454,7458,7459],[91,11365,11367],{"className":11366},[1209],[91,11368,11370,11413],{"className":11369},[154,155],[91,11371,11373,11410],{"className":11372},[159],[91,11374,11376,11396],{"className":11375,"style":7473},[163],[91,11377,11378,11381],{"style":7476},[91,11379],{"className":11380,"style":1226},[171],[91,11382,11384],{"className":11383},[176,177,178,179],[91,11385,11387,11390,11393],{"className":11386},[142,179],[91,11388,2249],{"className":11389},[142,212,179],[91,11391,332],{"className":11392},[425,179],[91,11394,571],{"className":11395},[142,179],[91,11397,11398,11401],{"style":7497},[91,11399],{"className":11400,"style":1226},[171],[91,11402,11404],{"className":11403},[176,177,178,179],[91,11405,11407],{"className":11406},[142,179],[91,11408,7371],{"className":11409},[142,179],[91,11411,218],{"className":11412},[217],[91,11414,11416],{"className":11415},[159],[91,11417,11419],{"className":11418,"style":7519},[163],[91,11420],{},[91,11422],{"className":11423,"style":1504},[420],[91,11425,11427,11430,11587],{"className":11426},[142],[91,11428],{"className":11429},[146,147],[91,11431,11433],{"className":11432},[110],[91,11434,11436,11579],{"className":11435},[154,155],[91,11437,11439,11576],{"className":11438},[159],[91,11440,11442,11482,11490],{"className":11441,"style":10300},[163],[91,11443,11444,11447],{"style":167},[91,11445],{"className":11446,"style":172},[171],[91,11448,11450],{"className":11449},[176,177,178,179],[91,11451,11453],{"className":11452},[142,179],[91,11454,11456,11459],{"className":11455},[142,179],[91,11457,119],{"className":11458},[142,179],[91,11460,11462],{"className":11461},[1209],[91,11463,11465],{"className":11464},[154],[91,11466,11468],{"className":11467},[159],[91,11469,11471],{"className":11470,"style":11077},[163],[91,11472,11473,11476],{"style":7576},[91,11474],{"className":11475,"style":7580},[171],[91,11477,11479],{"className":11478},[176,7584,1131,179],[91,11480,2249],{"className":11481},[142,212,179],[91,11483,11484,11487],{"style":188},[91,11485],{"className":11486,"style":172},[171],[91,11488],{"className":11489,"style":196},[195],[91,11491,11492,11495],{"style":10334},[91,11493],{"className":11494,"style":172},[171],[91,11496,11498],{"className":11497},[176,177,178,179],[91,11499,11501],{"className":11500},[142,179],[91,11502,11504,11513,11567],{"className":11503},[142,179],[91,11505,11507],{"className":11506,"style":10351},[146,176,7584,10350,179,1125],[91,11508,11510],{"className":11509},[1130,1131,179],[91,11511,1081],{"className":11512},[179],[91,11514,11516],{"className":11515},[110],[91,11517,11519,11559],{"className":11518},[154,155],[91,11520,11522,11556],{"className":11521},[159],[91,11523,11525,11539],{"className":11524,"style":10370},[163],[91,11526,11527,11530],{"style":10373},[91,11528],{"className":11529,"style":7580},[171],[91,11531,11533],{"className":11532},[176,7584,1131,179],[91,11534,11536],{"className":11535},[142,179],[91,11537,2249],{"className":11538},[142,212,179],[91,11540,11541,11544],{"style":10388},[91,11542],{"className":11543,"style":7580},[171],[91,11545,11547],{"className":11546},[176,7584,1131,179],[91,11548,11550,11553],{"className":11549},[142,179],[91,11551,458],{"className":11552},[142,179],[91,11554,2249],{"className":11555},[142,212,179],[91,11557,218],{"className":11558},[217],[91,11560,11562],{"className":11561},[159],[91,11563,11565],{"className":11564,"style":10413},[163],[91,11566],{},[91,11568,11570],{"className":11569,"style":10351},[231,176,7584,10350,179,1125],[91,11571,11573],{"className":11572},[1130,1131,179],[91,11574,1090],{"className":11575},[179],[91,11577,218],{"className":11578},[217],[91,11580,11582],{"className":11581},[159],[91,11583,11585],{"className":11584,"style":225},[163],[91,11586],{},[91,11588],{"className":11589},[231,147],[91,11591],{"className":11592,"style":2275},[420],[91,11594,7396],{"className":11595},[2279],[91,11597],{"className":11598,"style":2275},[420],[91,11600,11602,11606],{"className":11601},[133],[91,11603],{"className":11604,"style":11605},[137],"height:1.4213em;vertical-align:-0.4033em;",[91,11607,11609,11612,11721],{"className":11608},[142],[91,11610],{"className":11611},[146,147],[91,11613,11615],{"className":11614},[110],[91,11616,11618,11713],{"className":11617},[154,155],[91,11619,11621,11710],{"className":11620},[159],[91,11622,11625,11648,11656],{"className":11623,"style":11624},[163],"height:1.0179em;",[91,11626,11627,11630],{"style":167},[91,11628],{"className":11629,"style":172},[171],[91,11631,11633],{"className":11632},[176,177,178,179],[91,11634,11636,11639,11642,11645],{"className":11635},[142,179],[91,11637,458],{"className":11638},[142,179],[91,11640,2249],{"className":11641},[142,212,179],[91,11643,1368],{"className":11644},[2279,179],[91,11646,326],{"className":11647},[142,179],[91,11649,11650,11653],{"style":188},[91,11651],{"className":11652,"style":172},[171],[91,11654],{"className":11655,"style":196},[195],[91,11657,11658,11661],{"style":199},[91,11659],{"className":11660,"style":172},[171],[91,11662,11664],{"className":11663},[176,177,178,179],[91,11665,11667],{"className":11666},[142,179],[91,11668,11670,11673],{"className":11669},[142,179],[91,11671,11247],{"className":11672},[142,212,179],[91,11674,11676],{"className":11675},[1209],[91,11677,11679],{"className":11678},[154],[91,11680,11682],{"className":11681},[159],[91,11683,11686],{"className":11684,"style":11685},[163],"height:0.8913em;",[91,11687,11689,11692],{"style":11688},"top:-2.931em;margin-right:0.0714em;",[91,11690],{"className":11691,"style":7580},[171],[91,11693,11695],{"className":11694},[176,7584,1131,179],[91,11696,11698,11701,11704,11707],{"className":11697},[142,179],[91,11699,458],{"className":11700},[142,179],[91,11702,2249],{"className":11703},[142,212,179],[91,11705,1368],{"className":11706},[2279,179],[91,11708,326],{"className":11709},[142,179],[91,11711,218],{"className":11712},[217],[91,11714,11716],{"className":11715},[159],[91,11717,11719],{"className":11718,"style":3788},[163],[91,11720],{},[91,11722],{"className":11723},[231,147],[17,11725,11726,11727,11755,11756,11784,11785,40],{},"¿Te parece familiar? Es exactamente nuestra serie, multiplicada por ",[91,11728,11730,11743],{"className":11729},[94],[91,11731,11733],{"className":11732},[98],[100,11734,11735],{"xmlns":102},[104,11736,11737,11741],{},[107,11738,11739],{},[117,11740,458],{},[121,11742,458],{"encoding":123},[91,11744,11746],{"className":11745,"ariaHidden":129},[128],[91,11747,11749,11752],{"className":11748},[133],[91,11750],{"className":11751,"style":435},[137],[91,11753,458],{"className":11754},[142]," y con ",[91,11757,11759,11772],{"className":11758},[94],[91,11760,11762],{"className":11761},[98],[100,11763,11764],{"xmlns":102},[104,11765,11766,11770],{},[107,11767,11768],{},[113,11769,11247],{},[121,11771,11247],{"encoding":123},[91,11773,11775],{"className":11774,"ariaHidden":129},[128],[91,11776,11778,11781],{"className":11777},[133],[91,11779],{"className":11780,"style":261},[137],[91,11782,11247],{"className":11783},[142,212]," en lugar de ",[91,11786,11788,11801],{"className":11787},[94],[91,11789,11791],{"className":11790},[98],[100,11792,11793],{"xmlns":102},[104,11794,11795,11799],{},[107,11796,11797],{},[117,11798,326],{},[121,11800,326],{"encoding":123},[91,11802,11804],{"className":11803,"ariaHidden":129},[128],[91,11805,11807,11810],{"className":11806},[133],[91,11808],{"className":11809,"style":435},[137],[91,11811,326],{"className":11812},[142],[17,11814,11815,11816,11867],{},"Si evaluamos en ",[91,11817,11819,11837],{"className":11818},[94],[91,11820,11822],{"className":11821},[98],[100,11823,11824],{"xmlns":102},[104,11825,11826,11834],{},[107,11827,11828,11830,11832],{},[113,11829,11247],{},[330,11831,332],{},[117,11833,326],{},[121,11835,11836],{"encoding":123},"x = 1",[91,11838,11840,11858],{"className":11839,"ariaHidden":129},[128],[91,11841,11843,11846,11849,11852,11855],{"className":11842},[133],[91,11844],{"className":11845,"style":261},[137],[91,11847,11247],{"className":11848},[142,212],[91,11850],{"className":11851,"style":421},[420],[91,11853,332],{"className":11854},[425],[91,11856],{"className":11857,"style":421},[420],[91,11859,11861,11864],{"className":11860},[133],[91,11862],{"className":11863,"style":435},[137],[91,11865,326],{"className":11866},[142],":",[17,11869,11871],{"className":11870},[645],[91,11872,11874,11950],{"className":11873},[94],[91,11875,11877],{"className":11876},[98],[100,11878,11879],{"xmlns":102},[104,11880,11881,11947],{},[107,11882,11883,11885,11887,11889,11891,11893,11895,11909],{},[113,11884,11239],{},[330,11886,11242],{},[330,11888,1081],{"stretchy":9868},[117,11890,326],{},[330,11892,1090],{"stretchy":9868},[330,11894,332],{},[7355,11896,11897,11899,11907],{},[330,11898,7359],{},[107,11900,11901,11903,11905],{},[113,11902,2249],{},[330,11904,332],{},[117,11906,571],{},[113,11908,7371],{"mathvariant":7370},[110,11910,11911,11927],{},[107,11912,11913,11915,11925],{},[330,11914,1081],{"fence":129},[110,11916,11917,11923],{"linethickness":4373},[107,11918,11919,11921],{},[117,11920,458],{},[113,11922,2249],{},[113,11924,2249],{},[330,11926,1090],{"fence":129},[107,11928,11929,11935,11937,11939,11941,11943,11945],{},[1075,11930,11931,11933],{},[117,11932,119],{},[113,11934,2249],{},[330,11936,1081],{"stretchy":9868},[117,11938,458],{},[113,11940,2249],{},[330,11942,1368],{},[117,11944,326],{},[330,11946,1090],{"stretchy":9868},[121,11948,11949],{"encoding":123},"\\arcsin(1) = \\sum_{n=0}^{\\infty} \\frac{\\binom{2n}{n}}{4^n(2n+1)}",[91,11951,11953,11980],{"className":11952,"ariaHidden":129},[128],[91,11954,11956,11959,11962,11965,11968,11971,11974,11977],{"className":11955},[133],[91,11957],{"className":11958,"style":9889},[137],[91,11960,11239],{"className":11961},[7454],[91,11963,1081],{"className":11964},[146],[91,11966,326],{"className":11967},[142],[91,11969,1090],{"className":11970},[231],[91,11972],{"className":11973,"style":421},[420],[91,11975,332],{"className":11976},[425],[91,11978],{"className":11979,"style":421},[420],[91,11981,11983,11986,12049,12052],{"className":11982},[133],[91,11984],{"className":11985,"style":10614},[137],[91,11987,11989,11992],{"className":11988},[7454],[91,11990,7359],{"className":11991,"style":7460},[7454,7458,7459],[91,11993,11995],{"className":11994},[1209],[91,11996,11998,12041],{"className":11997},[154,155],[91,11999,12001,12038],{"className":12000},[159],[91,12002,12004,12024],{"className":12003,"style":7473},[163],[91,12005,12006,12009],{"style":7476},[91,12007],{"className":12008,"style":1226},[171],[91,12010,12012],{"className":12011},[176,177,178,179],[91,12013,12015,12018,12021],{"className":12014},[142,179],[91,12016,2249],{"className":12017},[142,212,179],[91,12019,332],{"className":12020},[425,179],[91,12022,571],{"className":12023},[142,179],[91,12025,12026,12029],{"style":7497},[91,12027],{"className":12028,"style":1226},[171],[91,12030,12032],{"className":12031},[176,177,178,179],[91,12033,12035],{"className":12034},[142,179],[91,12036,7371],{"className":12037},[142,179],[91,12039,218],{"className":12040},[217],[91,12042,12044],{"className":12043},[159],[91,12045,12047],{"className":12046,"style":7519},[163],[91,12048],{},[91,12050],{"className":12051,"style":1504},[420],[91,12053,12055,12058,12233],{"className":12054},[142],[91,12056],{"className":12057},[146,147],[91,12059,12061],{"className":12060},[110],[91,12062,12064,12225],{"className":12063},[154,155],[91,12065,12067,12222],{"className":12066},[159],[91,12068,12070,12128,12136],{"className":12069,"style":10300},[163],[91,12071,12072,12075],{"style":167},[91,12073],{"className":12074,"style":172},[171],[91,12076,12078],{"className":12077},[176,177,178,179],[91,12079,12081,12110,12113,12116,12119,12122,12125],{"className":12080},[142,179],[91,12082,12084,12087],{"className":12083},[142,179],[91,12085,119],{"className":12086},[142,179],[91,12088,12090],{"className":12089},[1209],[91,12091,12093],{"className":12092},[154],[91,12094,12096],{"className":12095},[159],[91,12097,12099],{"className":12098,"style":11077},[163],[91,12100,12101,12104],{"style":7576},[91,12102],{"className":12103,"style":7580},[171],[91,12105,12107],{"className":12106},[176,7584,1131,179],[91,12108,2249],{"className":12109},[142,212,179],[91,12111,1081],{"className":12112},[146,179],[91,12114,458],{"className":12115},[142,179],[91,12117,2249],{"className":12118},[142,212,179],[91,12120,1368],{"className":12121},[2279,179],[91,12123,326],{"className":12124},[142,179],[91,12126,1090],{"className":12127},[231,179],[91,12129,12130,12133],{"style":188},[91,12131],{"className":12132,"style":172},[171],[91,12134],{"className":12135,"style":196},[195],[91,12137,12138,12141],{"style":10334},[91,12139],{"className":12140,"style":172},[171],[91,12142,12144],{"className":12143},[176,177,178,179],[91,12145,12147],{"className":12146},[142,179],[91,12148,12150,12159,12213],{"className":12149},[142,179],[91,12151,12153],{"className":12152,"style":10351},[146,176,7584,10350,179,1125],[91,12154,12156],{"className":12155},[1130,1131,179],[91,12157,1081],{"className":12158},[179],[91,12160,12162],{"className":12161},[110],[91,12163,12165,12205],{"className":12164},[154,155],[91,12166,12168,12202],{"className":12167},[159],[91,12169,12171,12185],{"className":12170,"style":10370},[163],[91,12172,12173,12176],{"style":10373},[91,12174],{"className":12175,"style":7580},[171],[91,12177,12179],{"className":12178},[176,7584,1131,179],[91,12180,12182],{"className":12181},[142,179],[91,12183,2249],{"className":12184},[142,212,179],[91,12186,12187,12190],{"style":10388},[91,12188],{"className":12189,"style":7580},[171],[91,12191,12193],{"className":12192},[176,7584,1131,179],[91,12194,12196,12199],{"className":12195},[142,179],[91,12197,458],{"className":12198},[142,179],[91,12200,2249],{"className":12201},[142,212,179],[91,12203,218],{"className":12204},[217],[91,12206,12208],{"className":12207},[159],[91,12209,12211],{"className":12210,"style":10413},[163],[91,12212],{},[91,12214,12216],{"className":12215,"style":10351},[231,176,7584,10350,179,1125],[91,12217,12219],{"className":12218},[1130,1131,179],[91,12220,1090],{"className":12221},[179],[91,12223,218],{"className":12224},[217],[91,12226,12228],{"className":12227},[159],[91,12229,12231],{"className":12230,"style":10872},[163],[91,12232],{},[91,12234],{"className":12235},[231,147],[17,12237,12238,12239,12285,12286,12314,12315,11867],{},"¿Y cuánto vale ",[91,12240,12242,12264],{"className":12241},[94],[91,12243,12245],{"className":12244},[98],[100,12246,12247],{"xmlns":102},[104,12248,12249,12261],{},[107,12250,12251,12253,12255,12257,12259],{},[113,12252,11239],{},[330,12254,11242],{},[330,12256,1081],{"stretchy":9868},[117,12258,326],{},[330,12260,1090],{"stretchy":9868},[121,12262,12263],{"encoding":123},"\\arcsin(1)",[91,12265,12267],{"className":12266,"ariaHidden":129},[128],[91,12268,12270,12273,12276,12279,12282],{"className":12269},[133],[91,12271],{"className":12272,"style":9889},[137],[91,12274,11239],{"className":12275},[7454],[91,12277,1081],{"className":12278},[146],[91,12280,326],{"className":12281},[142],[91,12283,1090],{"className":12284},[231],"? Es el ángulo cuyo seno es ",[91,12287,12289,12302],{"className":12288},[94],[91,12290,12292],{"className":12291},[98],[100,12293,12294],{"xmlns":102},[104,12295,12296,12300],{},[107,12297,12298],{},[117,12299,326],{},[121,12301,326],{"encoding":123},[91,12303,12305],{"className":12304,"ariaHidden":129},[128],[91,12306,12308,12311],{"className":12307},[133],[91,12309],{"className":12310,"style":435},[137],[91,12312,326],{"className":12313},[142],", es decir, ",[91,12316,12318,12336],{"className":12317},[94],[91,12319,12321],{"className":12320},[98],[100,12322,12323],{"xmlns":102},[104,12324,12325,12333],{},[107,12326,12327,12330],{},[117,12328,12329],{},"90",[113,12331,12332],{"mathvariant":7370},"°",[121,12334,12335],{"encoding":123},"90°",[91,12337,12339],{"className":12338,"ariaHidden":129},[128],[91,12340,12342,12345],{"className":12341},[133],[91,12343],{"className":12344,"style":7078},[137],[91,12346,12335],{"className":12347},[142],[17,12349,12351],{"className":12350},[645],[91,12352,12354,12384],{"className":12353},[94],[91,12355,12357],{"className":12356},[98],[100,12358,12359],{"xmlns":102},[104,12360,12361,12381],{},[107,12362,12363,12365,12367,12369,12371,12373,12375],{},[113,12364,11239],{},[330,12366,11242],{},[330,12368,1081],{"stretchy":9868},[117,12370,326],{},[330,12372,1090],{"stretchy":9868},[330,12374,332],{},[110,12376,12377,12379],{},[113,12378,115],{},[117,12380,458],{},[121,12382,12383],{"encoding":123},"\\arcsin(1) = \\frac{\\pi}{2}",[91,12385,12387,12414],{"className":12386,"ariaHidden":129},[128],[91,12388,12390,12393,12396,12399,12402,12405,12408,12411],{"className":12389},[133],[91,12391],{"className":12392,"style":9889},[137],[91,12394,11239],{"className":12395},[7454],[91,12397,1081],{"className":12398},[146],[91,12400,326],{"className":12401},[142],[91,12403,1090],{"className":12404},[231],[91,12406],{"className":12407,"style":421},[420],[91,12409,332],{"className":12410},[425],[91,12412],{"className":12413,"style":421},[420],[91,12415,12417,12420],{"className":12416},[133],[91,12418],{"className":12419,"style":138},[137],[91,12421,12423,12426,12485],{"className":12422},[142],[91,12424],{"className":12425},[146,147],[91,12427,12429],{"className":12428},[110],[91,12430,12432,12477],{"className":12431},[154,155],[91,12433,12435,12474],{"className":12434},[159],[91,12436,12438,12452,12460],{"className":12437,"style":164},[163],[91,12439,12440,12443],{"style":167},[91,12441],{"className":12442,"style":172},[171],[91,12444,12446],{"className":12445},[176,177,178,179],[91,12447,12449],{"className":12448},[142,179],[91,12450,458],{"className":12451},[142,179],[91,12453,12454,12457],{"style":188},[91,12455],{"className":12456,"style":172},[171],[91,12458],{"className":12459,"style":196},[195],[91,12461,12462,12465],{"style":199},[91,12463],{"className":12464,"style":172},[171],[91,12466,12468],{"className":12467},[176,177,178,179],[91,12469,12471],{"className":12470},[142,179],[91,12472,115],{"className":12473,"style":213},[142,212,179],[91,12475,218],{"className":12476},[217],[91,12478,12480],{"className":12479},[159],[91,12481,12483],{"className":12482,"style":225},[163],[91,12484],{},[91,12486],{"className":12487},[231,147],[17,12489,12490],{},"Por lo tanto:",[17,12492,12494],{"className":12493},[645],[91,12495,12497,12535],{"className":12496},[94],[91,12498,12500],{"className":12499},[98],[100,12501,12502],{"xmlns":102},[104,12503,12504,12532],{},[107,12505,12506,12508,12510,12516,12518,12524,12526],{},[659,12507,7351],{},[330,12509,332],{},[110,12511,12512,12514],{},[117,12513,326],{},[117,12515,458],{},[330,12517,7396],{},[110,12519,12520,12522],{},[113,12521,115],{},[117,12523,458],{},[330,12525,332],{},[110,12527,12528,12530],{},[113,12529,115],{},[117,12531,119],{},[121,12533,12534],{"encoding":123},"\\text{Media} = \\frac{1}{2} \\cdot \\frac{\\pi}{2} = \\frac{\\pi}{4}",[91,12536,12538,12559,12642,12725],{"className":12537,"ariaHidden":129},[128],[91,12539,12541,12544,12550,12553,12556],{"className":12540},[133],[91,12542],{"className":12543,"style":7078},[137],[91,12545,12547],{"className":12546},[142,678],[91,12548,7351],{"className":12549},[142],[91,12551],{"className":12552,"style":421},[420],[91,12554,332],{"className":12555},[425],[91,12557],{"className":12558,"style":421},[420],[91,12560,12562,12565,12633,12636,12639],{"className":12561},[133],[91,12563],{"className":12564,"style":347},[137],[91,12566,12568,12571,12630],{"className":12567},[142],[91,12569],{"className":12570},[146,147],[91,12572,12574],{"className":12573},[110],[91,12575,12577,12622],{"className":12576},[154,155],[91,12578,12580,12619],{"className":12579},[159],[91,12581,12583,12597,12605],{"className":12582,"style":366},[163],[91,12584,12585,12588],{"style":167},[91,12586],{"className":12587,"style":172},[171],[91,12589,12591],{"className":12590},[176,177,178,179],[91,12592,12594],{"className":12593},[142,179],[91,12595,458],{"className":12596},[142,179],[91,12598,12599,12602],{"style":188},[91,12600],{"className":12601,"style":172},[171],[91,12603],{"className":12604,"style":196},[195],[91,12606,12607,12610],{"style":199},[91,12608],{"className":12609,"style":172},[171],[91,12611,12613],{"className":12612},[176,177,178,179],[91,12614,12616],{"className":12615},[142,179],[91,12617,326],{"className":12618},[142,179],[91,12620,218],{"className":12621},[217],[91,12623,12625],{"className":12624},[159],[91,12626,12628],{"className":12627,"style":225},[163],[91,12629],{},[91,12631],{"className":12632},[231,147],[91,12634],{"className":12635,"style":2275},[420],[91,12637,7396],{"className":12638},[2279],[91,12640],{"className":12641,"style":2275},[420],[91,12643,12645,12648,12716,12719,12722],{"className":12644},[133],[91,12646],{"className":12647,"style":138},[137],[91,12649,12651,12654,12713],{"className":12650},[142],[91,12652],{"className":12653},[146,147],[91,12655,12657],{"className":12656},[110],[91,12658,12660,12705],{"className":12659},[154,155],[91,12661,12663,12702],{"className":12662},[159],[91,12664,12666,12680,12688],{"className":12665,"style":164},[163],[91,12667,12668,12671],{"style":167},[91,12669],{"className":12670,"style":172},[171],[91,12672,12674],{"className":12673},[176,177,178,179],[91,12675,12677],{"className":12676},[142,179],[91,12678,458],{"className":12679},[142,179],[91,12681,12682,12685],{"style":188},[91,12683],{"className":12684,"style":172},[171],[91,12686],{"className":12687,"style":196},[195],[91,12689,12690,12693],{"style":199},[91,12691],{"className":12692,"style":172},[171],[91,12694,12696],{"className":12695},[176,177,178,179],[91,12697,12699],{"className":12698},[142,179],[91,12700,115],{"className":12701,"style":213},[142,212,179],[91,12703,218],{"className":12704},[217],[91,12706,12708],{"className":12707},[159],[91,12709,12711],{"className":12710,"style":225},[163],[91,12712],{},[91,12714],{"className":12715},[231,147],[91,12717],{"className":12718,"style":421},[420],[91,12720,332],{"className":12721},[425],[91,12723],{"className":12724,"style":421},[420],[91,12726,12728,12731],{"className":12727},[133],[91,12729],{"className":12730,"style":138},[137],[91,12732,12734,12737,12796],{"className":12733},[142],[91,12735],{"className":12736},[146,147],[91,12738,12740],{"className":12739},[110],[91,12741,12743,12788],{"className":12742},[154,155],[91,12744,12746,12785],{"className":12745},[159],[91,12747,12749,12763,12771],{"className":12748,"style":164},[163],[91,12750,12751,12754],{"style":167},[91,12752],{"className":12753,"style":172},[171],[91,12755,12757],{"className":12756},[176,177,178,179],[91,12758,12760],{"className":12759},[142,179],[91,12761,119],{"className":12762},[142,179],[91,12764,12765,12768],{"style":188},[91,12766],{"className":12767,"style":172},[171],[91,12769],{"className":12770,"style":196},[195],[91,12772,12773,12776],{"style":199},[91,12774],{"className":12775,"style":172},[171],[91,12777,12779],{"className":12778},[176,177,178,179],[91,12780,12782],{"className":12781},[142,179],[91,12783,115],{"className":12784,"style":213},[142,212,179],[91,12786,218],{"className":12787},[217],[91,12789,12791],{"className":12790},[159],[91,12792,12794],{"className":12793,"style":225},[163],[91,12795],{},[91,12797],{"className":12798},[231,147],[17,12800,12801,12802,12810,12811,12839,12840,12843],{},"Esta conexión fue ",[24,12803,12806,12807],{"href":12804,"rel":12805},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2602.14487",[81],"descubierta en 2025 por el matemático ",[33,12808,12809],{},"Jim Propp",". Y lo que la hace extraordinaria no es que ",[91,12812,12814,12827],{"className":12813},[94],[91,12815,12817],{"className":12816},[98],[100,12818,12819],{"xmlns":102},[104,12820,12821,12825],{},[107,12822,12823],{},[113,12824,115],{},[121,12826,251],{"encoding":123},[91,12828,12830],{"className":12829,"ariaHidden":129},[128],[91,12831,12833,12836],{"className":12832},[133],[91,12834],{"className":12835,"style":261},[137],[91,12837,115],{"className":12838,"style":213},[142,212]," aparezca en una serie infinita — eso es habitual. Lo extraordinario es que una serie nacida de un proceso tan elemental como lanzar monedas resulte ser ",[267,12841,12842],{},"exactamente"," la misma serie que define una función trigonométrica.",[17,12845,12846],{},"No es una aproximación ni una coincidencia numérica — es una identidad matemática exacta.",[17,12848,12849,12850,12878,12879,11867],{},"Eso sí, no intentes calcular ",[91,12851,12853,12866],{"className":12852},[94],[91,12854,12856],{"className":12855},[98],[100,12857,12858],{"xmlns":102},[104,12859,12860,12864],{},[107,12861,12862],{},[113,12863,115],{},[121,12865,251],{"encoding":123},[91,12867,12869],{"className":12868,"ariaHidden":129},[128],[91,12870,12872,12875],{"className":12871},[133],[91,12873],{"className":12874,"style":261},[137],[91,12876,115],{"className":12877,"style":213},[142,212]," de esta forma. Como el propio Propp ",[24,12880,12882],{"href":12804,"rel":12881},[81],"explicó en el propio documento de Arxiv",[12884,12885,12886],"blockquote",{},[17,12887,12888],{},"Para obtener pi con la precisión de 3,14, podrían hacer falta hasta un billón de lanzamientos de moneda. En parte porque las secuencias pueden alargarse mucho antes de que las caras superen a las cruces — tanto que el valor esperado de la longitud de una secuencia es ¡infinito! Además, no se pueden lanzar todas las monedas a la vez como se pueden soltar agujas: el orden de caras y cruces importa.",[12,12890,12892],{"id":12891},"reflexión-final","Reflexión final",[17,12894,12895],{},"En el artículo anterior, descubrimos que el triángulo de Tartaglia conectaba la combinatoria, el álgebra y la probabilidad. Tres lenguajes que contaban la misma historia.",[17,12897,12898,12899,40],{},"Hoy hemos añadido un cuarto personaje a la trama: el número ",[91,12900,12902,12915],{"className":12901},[94],[91,12903,12905],{"className":12904},[98],[100,12906,12907],{"xmlns":102},[104,12908,12909,12913],{},[107,12910,12911],{},[113,12912,115],{},[121,12914,251],{"encoding":123},[91,12916,12918],{"className":12917,"ariaHidden":129},[128],[91,12919,12921,12924],{"className":12920},[133],[91,12922],{"className":12923,"style":261},[137],[91,12925,115],{"className":12926,"style":213},[142,212],[17,12928,12929,12930,12958],{},"Lanzar una moneda parece un proceso puramente aleatorio. El triángulo de Tartaglia parece pura aritmética. Los números de Catalan parecen un capricho de la combinatoria. Y ",[91,12931,12933,12946],{"className":12932},[94],[91,12934,12936],{"className":12935},[98],[100,12937,12938],{"xmlns":102},[104,12939,12940,12944],{},[107,12941,12942],{},[113,12943,115],{},[121,12945,251],{"encoding":123},[91,12947,12949],{"className":12948,"ariaHidden":129},[128],[91,12950,12952,12955],{"className":12951},[133],[91,12953],{"className":12954,"style":261},[137],[91,12956,115],{"className":12957,"style":213},[142,212]," parece pertenecer exclusivamente a la geometría — a los círculos, a las curvas, al mundo continuo.",[17,12960,12961],{},"Y sin embargo, todos terminan conectados en una sola ecuación.",[17,12963,12964,12965,12993],{},"Hay algo profundamente satisfactorio en descubrir que un hecho nuevo sobre ",[91,12966,12968,12981],{"className":12967},[94],[91,12969,12971],{"className":12970},[98],[100,12972,12973],{"xmlns":102},[104,12974,12975,12979],{},[107,12976,12977],{},[113,12978,115],{},[121,12980,251],{"encoding":123},[91,12982,12984],{"className":12983,"ariaHidden":129},[128],[91,12985,12987,12990],{"className":12986},[133],[91,12988],{"className":12989,"style":261},[137],[91,12991,115],{"className":12992,"style":213},[142,212]," se escondía, desde siempre, detrás de algo tan mundano como lanzar una moneda. Y que nadie lo notó hasta 2025.",[17,12995,12996],{},"Las matemáticas siguen teniendo secretos. Y a veces, están más cerca de lo que pensamos.",[17,12998,12999,13000,13004],{},"Pon la teoría en práctica: ",[24,13001,13003],{"href":631,"rel":13002},[81],"lanza una moneda online"," y observa los patrones en acción.",{"title":51,"searchDepth":52,"depth":52,"links":13006},[13007,13015,13016,13017,13018,13019],{"id":616,"depth":52,"text":617,"children":13008},[13009,13011,13012,13013,13014],{"id":637,"depth":13010,"text":638},3,{"id":905,"depth":13010,"text":906},{"id":1319,"depth":13010,"text":1320},{"id":2206,"depth":13010,"text":2207},{"id":2217,"depth":13010,"text":2218},{"id":4126,"depth":52,"text":4127},{"id":6205,"depth":52,"text":6206},{"id":6411,"depth":52,"text":6412},{"id":9388,"depth":52,"text":9389},{"id":12891,"depth":52,"text":12892},"2026-05-27T10:00:00Z","Descubre cómo lanzar una moneda hasta que haya más caras que cruces conduce, inesperadamente, al número π — a través de los números de Catalan y la serie del arcoseno.",{"excerpt":13023},{"type":9,"value":13024},[13025,13027,13132,13164],[71,13026,67],{"id":73},[17,13028,76,13029,83,13032,89,13035,40],{},[24,13030,82],{"href":79,"rel":13031},[81],[24,13033,88],{"href":86,"rel":13034},[81],[91,13036,13038,13055],{"className":13037},[94],[91,13039,13041],{"className":13040},[98],[100,13042,13043],{"xmlns":102},[104,13044,13045,13053],{},[107,13046,13047],{},[110,13048,13049,13051],{},[113,13050,115],{},[117,13052,119],{},[121,13054,124],{"encoding":123},[91,13056,13058],{"className":13057,"ariaHidden":129},[128],[91,13059,13061,13064],{"className":13060},[133],[91,13062],{"className":13063,"style":138},[137],[91,13065,13067,13070,13129],{"className":13066},[142],[91,13068],{"className":13069},[146,147],[91,13071,13073],{"className":13072},[110],[91,13074,13076,13121],{"className":13075},[154,155],[91,13077,13079,13118],{"className":13078},[159],[91,13080,13082,13096,13104],{"className":13081,"style":164},[163],[91,13083,13084,13087],{"style":167},[91,13085],{"className":13086,"style":172},[171],[91,13088,13090],{"className":13089},[176,177,178,179],[91,13091,13093],{"className":13092},[142,179],[91,13094,119],{"className":13095},[142,179],[91,13097,13098,13101],{"style":188},[91,13099],{"className":13100,"style":172},[171],[91,13102],{"className":13103,"style":196},[195],[91,13105,13106,13109],{"style":199},[91,13107],{"className":13108,"style":172},[171],[91,13110,13112],{"className":13111},[176,177,178,179],[91,13113,13115],{"className":13114},[142,179],[91,13116,115],{"className":13117,"style":213},[142,212,179],[91,13119,218],{"className":13120},[217],[91,13122,13124],{"className":13123},[159],[91,13125,13127],{"className":13126,"style":225},[163],[91,13128],{},[91,13130],{"className":13131},[231,147],[17,13133,234,13134,265,13162,270],{},[91,13135,13137,13150],{"className":13136},[94],[91,13138,13140],{"className":13139},[98],[100,13141,13142],{"xmlns":102},[104,13143,13144,13148],{},[107,13145,13146],{},[113,13147,115],{},[121,13149,251],{"encoding":123},[91,13151,13153],{"className":13152,"ariaHidden":129},[128],[91,13154,13156,13159],{"className":13155},[133],[91,13157],{"className":13158,"style":261},[137],[91,13160,115],{"className":13161,"style":213},[142,212],[267,13163,269],{},[17,13165,273,13166,279],{},[24,13167,278],{"href":276,"rel":13168},[81],"\u002Fblog\u002Fpi-coin-flips-catalan-numbers.es",{"title":67,"description":13021},"lanzando-una-moneda-hasta-obtener-pi","blog\u002Fpi-coin-flips-catalan-numbers.es","tftvlIus-Vzedr65fsds1GlLWOwi3lEU2RboB8VhTN4",{"id":13175,"title":13176,"_locale":7,"body":13177,"date":23713,"description":23714,"extension":56,"meta":23715,"navigation":58,"path":23722,"published":58,"seo":23723,"slug":23724,"stem":23725,"__hash__":23726},"blog\u002Fblog\u002Fpascals-triangle-and-probability.es.md","¿Qué tienen en común el triángulo de Tartaglia y una moneda?",{"type":9,"value":13178,"toc":23698},[13179,13182,13185,13189,13204,13207,13213,13217,13220,13241,13244,13250,13260,13506,13510,13516,13623,13742,13898,13901,14147,14449,14892,14924,15323,15327,15337,15546,15552,15616,15676,15808,15811,15819,15823,15909,15915,15922,15975,16051,16269,16273,16278,16413,16445,16871,16874,17581,17767,17770,18041,18045,18048,18132,18216,18348,18354,18486,18489,19280,19283,19289,19292,19312,19320,19324,19331,19426,19429,19623,19655,19660,19890,19949,19954,20276,20346,20351,20857,20954,20964,21330,21566,21570,21634,21946,21949,22916,23291,23294,23560,23655,23657,23660,23677,23683,23686,23689,23692],[71,13180,13176],{"id":13181},"qué-tienen-en-común-el-triángulo-de-tartaglia-y-una-moneda",[17,13183,13184],{},"Existe un triángulo de números que lleva siglos fascinando a matemáticos y matemáticas de todo el mundo. Su construcción es tan sencilla que un niño o niña podría dibujarlo en un cuaderno. Y sin embargo, dentro de sus filas se esconde una conexión profunda entre tres ramas de las matemáticas que, a primera vista, no parecen tener nada que ver entre sí: la combinatoria, el álgebra y la probabilidad. Lo más sorprendente es que esa conexión se revela con algo tan cotidiano como lanzar una moneda.",[12,13186,13188],{"id":13187},"un-poco-de-historia","Un poco de historia",[17,13190,13191,13192,13195,13196,13199,13200,13203],{},"En muchos países de habla hispana este patrón se conoce como ",[33,13193,13194],{},"triángulo de Tartaglia",", en honor al matemático italiano Niccolò Fontana Tartaglia (1499–1557), que lo estudió en el contexto de la resolución de ecuaciones cúbicas. En el mundo anglosajón se le llama ",[33,13197,13198],{},"triángulo de Pascal",", porque Blaise Pascal publicó en 1654 un tratado — el ",[267,13201,13202],{},"Traité du triangle arithmétique"," — donde formalizó sus propiedades y, lo que es más importante, lo conectó por primera vez con los problemas de azar que le planteaba el caballero de Méré.",[17,13205,13206],{},"Pero la historia real de este triángulo es mucho más antigua. En la China del siglo XI, el matemático Jia Xian ya describía el mismo patrón para calcular coeficientes de potencias. Más tarde, en el siglo XIII, Yang Hui lo reprodujo en un diagrama que ha llegado hasta nuestros días. En Persia, Omar Jayyam trabajó con las mismas ideas en el siglo XII. Y en la India, Pingala exploró patrones combinatorios similares siglos antes de nuestra era.",[17,13208,13209,13210],{},"Que civilizaciones tan distantes en el tiempo y el espacio descubrieran el mismo triángulo de forma independiente nos dice algo importante: no se trata de una invención humana arbitraria, sino de una estructura que emerge de forma natural cada vez que alguien se pregunta ",[267,13211,13212],{},"¿de cuántas maneras puedo organizar esto?",[12,13214,13216],{"id":13215},"cómo-construir-el-triángulo","Cómo construir el triángulo",[17,13218,13219],{},"La regla de construcción es elegantemente simple:",[284,13221,13222,13228,13234],{},[287,13223,13224,13225,13227],{},"Empieza colocando un ",[33,13226,326],{}," en la cima.",[287,13229,13230,13231,13233],{},"En cada fila nueva, coloca un ",[33,13232,326],{}," en cada extremo.",[287,13235,13236,13237,13240],{},"Cada número interior es la ",[33,13238,13239],{},"suma de los dos números"," que tiene justo encima.",[17,13242,13243],{},"Las primeras filas quedan así:",[1339,13245,13248],{"className":13246,"code":13247,"language":678,"meta":51},[1342],"            1\n          1   1\n        1   2   1\n      1   3   3   1\n    1   4   6   4   1\n  1   5  10  10   5   1\n",[1345,13249,13247],{"__ignoreMap":51},[17,13251,13252,13253,13255,13256,13259],{},"La primera fila (el ",[33,13254,326],{}," solitario de la cima) se llama ",[33,13257,13258],{},"fila 0",". La siguiente es la fila 1, y así sucesivamente. Esto no es un capricho: numerar desde cero hará que todo encaje mejor cuando conectemos el triángulo con las fórmulas.",[17,13261,13262,13263,13344,13345,13373,13374,13344,13425,13453,13454,13505],{},"Fíjate en la fila 4: ",[91,13264,13266,13296],{"className":13265},[94],[91,13267,13269],{"className":13268},[98],[100,13270,13271],{"xmlns":102},[104,13272,13273,13293],{},[107,13274,13275,13277,13279,13281,13283,13285,13287,13289,13291],{},[117,13276,326],{},[330,13278,579],{"separator":129},[117,13280,119],{},[330,13282,579],{"separator":129},[117,13284,5494],{},[330,13286,579],{"separator":129},[117,13288,119],{},[330,13290,579],{"separator":129},[117,13292,326],{},[121,13294,13295],{"encoding":123},"1, 4, 6, 4, 1",[91,13297,13299],{"className":13298,"ariaHidden":129},[128],[91,13300,13302,13305,13308,13311,13314,13317,13320,13323,13326,13329,13332,13335,13338,13341],{"className":13301},[133],[91,13303],{"className":13304,"style":567},[137],[91,13306,326],{"className":13307},[142],[91,13309,579],{"className":13310},[578],[91,13312],{"className":13313,"style":1504},[420],[91,13315,119],{"className":13316},[142],[91,13318,579],{"className":13319},[578],[91,13321],{"className":13322,"style":1504},[420],[91,13324,5494],{"className":13325},[142],[91,13327,579],{"className":13328},[578],[91,13330],{"className":13331,"style":1504},[420],[91,13333,119],{"className":13334},[142],[91,13336,579],{"className":13337},[578],[91,13339],{"className":13340,"style":1504},[420],[91,13342,326],{"className":13343},[142],". El ",[91,13346,13348,13361],{"className":13347},[94],[91,13349,13351],{"className":13350},[98],[100,13352,13353],{"xmlns":102},[104,13354,13355,13359],{},[107,13356,13357],{},[117,13358,5494],{},[121,13360,5494],{"encoding":123},[91,13362,13364],{"className":13363,"ariaHidden":129},[128],[91,13365,13367,13370],{"className":13366},[133],[91,13368],{"className":13369,"style":435},[137],[91,13371,5494],{"className":13372},[142]," del centro viene de sumar los dos números que tiene encima: ",[91,13375,13377,13395],{"className":13376},[94],[91,13378,13380],{"className":13379},[98],[100,13381,13382],{"xmlns":102},[104,13383,13384,13392],{},[107,13385,13386,13388,13390],{},[117,13387,980],{},[330,13389,1368],{},[117,13391,980],{},[121,13393,13394],{"encoding":123},"3 + 3",[91,13396,13398,13416],{"className":13397,"ariaHidden":129},[128],[91,13399,13401,13404,13407,13410,13413],{"className":13400},[133],[91,13402],{"className":13403,"style":1383},[137],[91,13405,980],{"className":13406},[142],[91,13408],{"className":13409,"style":2275},[420],[91,13411,1368],{"className":13412},[2279],[91,13414],{"className":13415,"style":2275},[420],[91,13417,13419,13422],{"className":13418},[133],[91,13420],{"className":13421,"style":435},[137],[91,13423,980],{"className":13424},[142],[91,13426,13428,13441],{"className":13427},[94],[91,13429,13431],{"className":13430},[98],[100,13432,13433],{"xmlns":102},[104,13434,13435,13439],{},[107,13436,13437],{},[117,13438,119],{},[121,13440,119],{"encoding":123},[91,13442,13444],{"className":13443,"ariaHidden":129},[128],[91,13445,13447,13450],{"className":13446},[133],[91,13448],{"className":13449,"style":435},[137],[91,13451,119],{"className":13452},[142]," de la izquierda viene de ",[91,13455,13457,13475],{"className":13456},[94],[91,13458,13460],{"className":13459},[98],[100,13461,13462],{"xmlns":102},[104,13463,13464,13472],{},[107,13465,13466,13468,13470],{},[117,13467,326],{},[330,13469,1368],{},[117,13471,980],{},[121,13473,13474],{"encoding":123},"1 + 3",[91,13476,13478,13496],{"className":13477,"ariaHidden":129},[128],[91,13479,13481,13484,13487,13490,13493],{"className":13480},[133],[91,13482],{"className":13483,"style":1383},[137],[91,13485,326],{"className":13486},[142],[91,13488],{"className":13489,"style":2275},[420],[91,13491,1368],{"className":13492},[2279],[91,13494],{"className":13495,"style":2275},[420],[91,13497,13499,13502],{"className":13498},[133],[91,13500],{"className":13501,"style":435},[137],[91,13503,980],{"className":13504},[142],". Siempre la misma regla.",[635,13507,13509],{"id":13508},"los-coeficientes-binomiales","Los coeficientes binomiales",[17,13511,13512,13513,13515],{},"Cada número del triángulo tiene un nombre formal: es un ",[33,13514,4781],{},". Se escribe así:",[17,13517,13519],{"className":13518},[645],[91,13520,13522,13545],{"className":13521},[94],[91,13523,13525],{"className":13524},[98],[100,13526,13527],{"xmlns":102},[104,13528,13529,13542],{},[107,13530,13531,13533,13540],{},[330,13532,1081],{"fence":129},[110,13534,13535,13537],{"linethickness":4373},[113,13536,2249],{},[113,13538,13539],{},"k",[330,13541,1090],{"fence":129},[121,13543,13544],{"encoding":123},"\\binom{n}{k}",[91,13546,13548],{"className":13547,"ariaHidden":129},[128],[91,13549,13551,13555],{"className":13550},[133],[91,13552],{"className":13553,"style":13554},[137],"height:1.2em;vertical-align:-0.35em;",[91,13556,13558,13564,13617],{"className":13557},[142],[91,13559,13561],{"className":13560,"style":1126},[146,1125],[91,13562,1081],{"className":13563},[1130,1131],[91,13565,13567],{"className":13566},[110],[91,13568,13570,13609],{"className":13569},[154,155],[91,13571,13573,13606],{"className":13572},[159],[91,13574,13577,13592],{"className":13575,"style":13576},[163],"height:0.7454em;",[91,13578,13579,13582],{"style":4555},[91,13580],{"className":13581,"style":1226},[171],[91,13583,13585],{"className":13584},[176,177,178,179],[91,13586,13588],{"className":13587},[142,179],[91,13589,13539],{"className":13590,"style":13591},[142,212,179],"margin-right:0.0315em;",[91,13593,13594,13597],{"style":4570},[91,13595],{"className":13596,"style":1226},[171],[91,13598,13600],{"className":13599},[176,177,178,179],[91,13601,13603],{"className":13602},[142,179],[91,13604,2249],{"className":13605},[142,212,179],[91,13607,218],{"className":13608},[217],[91,13610,13612],{"className":13611},[159],[91,13613,13615],{"className":13614,"style":225},[163],[91,13616],{},[91,13618,13620],{"className":13619,"style":1126},[231,1125],[91,13621,1090],{"className":13622},[1130,1131],[17,13624,13625,13626,13684,13685,4811,13713,13741],{},"y se lee \"",[33,13627,13628,4749,13656],{},[91,13629,13631,13644],{"className":13630},[94],[91,13632,13634],{"className":13633},[98],[100,13635,13636],{"xmlns":102},[104,13637,13638,13642],{},[107,13639,13640],{},[113,13641,2249],{},[121,13643,2249],{"encoding":123},[91,13645,13647],{"className":13646,"ariaHidden":129},[128],[91,13648,13650,13653],{"className":13649},[133],[91,13651],{"className":13652,"style":261},[137],[91,13654,2249],{"className":13655},[142,212],[91,13657,13659,13672],{"className":13658},[94],[91,13660,13662],{"className":13661},[98],[100,13663,13664],{"xmlns":102},[104,13665,13666,13670],{},[107,13667,13668],{},[113,13669,13539],{},[121,13671,13539],{"encoding":123},[91,13673,13675],{"className":13674,"ariaHidden":129},[128],[91,13676,13678,13681],{"className":13677},[133],[91,13679],{"className":13680,"style":7078},[137],[91,13682,13539],{"className":13683,"style":13591},[142,212],"\". Su significado es directo: representa el número de formas de elegir ",[91,13686,13688,13701],{"className":13687},[94],[91,13689,13691],{"className":13690},[98],[100,13692,13693],{"xmlns":102},[104,13694,13695,13699],{},[107,13696,13697],{},[113,13698,13539],{},[121,13700,13539],{"encoding":123},[91,13702,13704],{"className":13703,"ariaHidden":129},[128],[91,13705,13707,13710],{"className":13706},[133],[91,13708],{"className":13709,"style":7078},[137],[91,13711,13539],{"className":13712,"style":13591},[142,212],[91,13714,13716,13729],{"className":13715},[94],[91,13717,13719],{"className":13718},[98],[100,13720,13721],{"xmlns":102},[104,13722,13723,13727],{},[107,13724,13725],{},[113,13726,2249],{},[121,13728,2249],{"encoding":123},[91,13730,13732],{"className":13731,"ariaHidden":129},[128],[91,13733,13735,13738],{"className":13734},[133],[91,13736],{"className":13737,"style":261},[137],[91,13739,2249],{"className":13740},[142,212],", sin importar el orden.",[17,13743,13744,13745,13869,13870,40],{},"Por ejemplo, ",[91,13746,13748,13776],{"className":13747},[94],[91,13749,13751],{"className":13750},[98],[100,13752,13753],{"xmlns":102},[104,13754,13755,13773],{},[107,13756,13757,13769,13771],{},[107,13758,13759,13761,13767],{},[330,13760,1081],{"fence":129},[110,13762,13763,13765],{"linethickness":4373},[117,13764,119],{},[117,13766,458],{},[330,13768,1090],{"fence":129},[330,13770,332],{},[117,13772,5494],{},[121,13774,13775],{"encoding":123},"\\binom{4}{2} = 6",[91,13777,13779,13860],{"className":13778,"ariaHidden":129},[128],[91,13780,13782,13785,13851,13854,13857],{"className":13781},[133],[91,13783],{"className":13784,"style":4644},[137],[91,13786,13788,13794,13845],{"className":13787},[142],[91,13789,13791],{"className":13790,"style":1126},[146,1125],[91,13792,1081],{"className":13793},[1130,1131],[91,13795,13797],{"className":13796},[110],[91,13798,13800,13837],{"className":13799},[154,155],[91,13801,13803,13834],{"className":13802},[159],[91,13804,13806,13820],{"className":13805,"style":4552},[163],[91,13807,13808,13811],{"style":4555},[91,13809],{"className":13810,"style":1226},[171],[91,13812,13814],{"className":13813},[176,177,178,179],[91,13815,13817],{"className":13816},[142,179],[91,13818,458],{"className":13819},[142,179],[91,13821,13822,13825],{"style":4570},[91,13823],{"className":13824,"style":1226},[171],[91,13826,13828],{"className":13827},[176,177,178,179],[91,13829,13831],{"className":13830},[142,179],[91,13832,119],{"className":13833},[142,179],[91,13835,218],{"className":13836},[217],[91,13838,13840],{"className":13839},[159],[91,13841,13843],{"className":13842,"style":225},[163],[91,13844],{},[91,13846,13848],{"className":13847,"style":1126},[231,1125],[91,13849,1090],{"className":13850},[1130,1131],[91,13852],{"className":13853,"style":421},[420],[91,13855,332],{"className":13856},[425],[91,13858],{"className":13859,"style":421},[420],[91,13861,13863,13866],{"className":13862},[133],[91,13864],{"className":13865,"style":435},[137],[91,13867,5494],{"className":13868},[142]," significa que hay 6 formas de elegir 2 objetos de entre 4. Y efectivamente, si miras la fila 4 del triángulo, la posición 2 (contando desde 0) es precisamente ",[91,13871,13873,13886],{"className":13872},[94],[91,13874,13876],{"className":13875},[98],[100,13877,13878],{"xmlns":102},[104,13879,13880,13884],{},[107,13881,13882],{},[117,13883,5494],{},[121,13885,5494],{"encoding":123},[91,13887,13889],{"className":13888,"ariaHidden":129},[128],[91,13890,13892,13895],{"className":13891},[133],[91,13893],{"className":13894,"style":435},[137],[91,13896,5494],{"className":13897},[142],[17,13899,13900],{},"La fórmula para calcularlo es:",[17,13902,13904],{"className":13903},[645],[91,13905,13907,13962],{"className":13906},[94],[91,13908,13910],{"className":13909},[98],[100,13911,13912],{"xmlns":102},[104,13913,13914,13959],{},[107,13915,13916,13928,13930],{},[107,13917,13918,13920,13926],{},[330,13919,1081],{"fence":129},[110,13921,13922,13924],{"linethickness":4373},[113,13923,2249],{},[113,13925,13539],{},[330,13927,1090],{"fence":129},[330,13929,332],{},[110,13931,13932,13939],{},[107,13933,13934,13936],{},[113,13935,2249],{},[330,13937,13938],{"stretchy":9868},"!",[107,13940,13941,13943,13945,13947,13949,13951,13953,13955,13957],{},[113,13942,13539],{},[330,13944,13938],{"stretchy":9868},[330,13946,7396],{},[330,13948,1081],{"stretchy":9868},[113,13950,2249],{},[330,13952,1405],{},[113,13954,13539],{},[330,13956,1090],{"stretchy":9868},[330,13958,13938],{"stretchy":9868},[121,13960,13961],{"encoding":123},"\\binom{n}{k} = \\frac{n!}{k! \\cdot (n-k)!}",[91,13963,13965,14046],{"className":13964,"ariaHidden":129},[128],[91,13966,13968,13971,14037,14040,14043],{"className":13967},[133],[91,13969],{"className":13970,"style":13554},[137],[91,13972,13974,13980,14031],{"className":13973},[142],[91,13975,13977],{"className":13976,"style":1126},[146,1125],[91,13978,1081],{"className":13979},[1130,1131],[91,13981,13983],{"className":13982},[110],[91,13984,13986,14023],{"className":13985},[154,155],[91,13987,13989,14020],{"className":13988},[159],[91,13990,13992,14006],{"className":13991,"style":13576},[163],[91,13993,13994,13997],{"style":4555},[91,13995],{"className":13996,"style":1226},[171],[91,13998,14000],{"className":13999},[176,177,178,179],[91,14001,14003],{"className":14002},[142,179],[91,14004,13539],{"className":14005,"style":13591},[142,212,179],[91,14007,14008,14011],{"style":4570},[91,14009],{"className":14010,"style":1226},[171],[91,14012,14014],{"className":14013},[176,177,178,179],[91,14015,14017],{"className":14016},[142,179],[91,14018,2249],{"className":14019},[142,212,179],[91,14021,218],{"className":14022},[217],[91,14024,14026],{"className":14025},[159],[91,14027,14029],{"className":14028,"style":225},[163],[91,14030],{},[91,14032,14034],{"className":14033,"style":1126},[231,1125],[91,14035,1090],{"className":14036},[1130,1131],[91,14038],{"className":14039,"style":421},[420],[91,14041,332],{"className":14042},[425],[91,14044],{"className":14045,"style":421},[420],[91,14047,14049,14053],{"className":14048},[133],[91,14050],{"className":14051,"style":14052},[137],"height:1.4001em;vertical-align:-0.52em;",[91,14054,14056,14059,14144],{"className":14055},[142],[91,14057],{"className":14058},[146,147],[91,14060,14062],{"className":14061},[110],[91,14063,14065,14136],{"className":14064},[154,155],[91,14066,14068,14133],{"className":14067},[159],[91,14069,14072,14108,14116],{"className":14070,"style":14071},[163],"height:0.8801em;",[91,14073,14074,14077],{"style":167},[91,14075],{"className":14076,"style":172},[171],[91,14078,14080],{"className":14079},[176,177,178,179],[91,14081,14083,14086,14089,14092,14095,14098,14101,14104],{"className":14082},[142,179],[91,14084,13539],{"className":14085,"style":13591},[142,212,179],[91,14087,13938],{"className":14088},[231,179],[91,14090,7396],{"className":14091},[2279,179],[91,14093,1081],{"className":14094},[146,179],[91,14096,2249],{"className":14097},[142,212,179],[91,14099,1405],{"className":14100},[2279,179],[91,14102,13539],{"className":14103,"style":13591},[142,212,179],[91,14105,14107],{"className":14106},[231,179],")!",[91,14109,14110,14113],{"style":188},[91,14111],{"className":14112,"style":172},[171],[91,14114],{"className":14115,"style":196},[195],[91,14117,14118,14121],{"style":199},[91,14119],{"className":14120,"style":172},[171],[91,14122,14124],{"className":14123},[176,177,178,179],[91,14125,14127,14130],{"className":14126},[142,179],[91,14128,2249],{"className":14129},[142,212,179],[91,14131,13938],{"className":14132},[231,179],[91,14134,218],{"className":14135},[217],[91,14137,14139],{"className":14138},[159],[91,14140,14142],{"className":14141,"style":10872},[163],[91,14143],{},[91,14145],{"className":14146},[231,147],[17,14148,14149,14150,14184,14185,14188,14189,14217,14218,14246,14247,14392,14393,40],{},"Aquí, el símbolo ",[91,14151,14153,14169],{"className":14152},[94],[91,14154,14156],{"className":14155},[98],[100,14157,14158],{"xmlns":102},[104,14159,14160,14166],{},[107,14161,14162,14164],{},[113,14163,2249],{},[330,14165,13938],{"stretchy":9868},[121,14167,14168],{"encoding":123},"n!",[91,14170,14172],{"className":14171,"ariaHidden":129},[128],[91,14173,14175,14178,14181],{"className":14174},[133],[91,14176],{"className":14177,"style":7078},[137],[91,14179,2249],{"className":14180},[142,212],[91,14182,13938],{"className":14183},[231]," se llama ",[33,14186,14187],{},"factorial"," y significa multiplicar todos los números enteros desde ",[91,14190,14192,14205],{"className":14191},[94],[91,14193,14195],{"className":14194},[98],[100,14196,14197],{"xmlns":102},[104,14198,14199,14203],{},[107,14200,14201],{},[117,14202,326],{},[121,14204,326],{"encoding":123},[91,14206,14208],{"className":14207,"ariaHidden":129},[128],[91,14209,14211,14214],{"className":14210},[133],[91,14212],{"className":14213,"style":435},[137],[91,14215,326],{"className":14216},[142]," hasta ",[91,14219,14221,14234],{"className":14220},[94],[91,14222,14224],{"className":14223},[98],[100,14225,14226],{"xmlns":102},[104,14227,14228,14232],{},[107,14229,14230],{},[113,14231,2249],{},[121,14233,2249],{"encoding":123},[91,14235,14237],{"className":14236,"ariaHidden":129},[128],[91,14238,14240,14243],{"className":14239},[133],[91,14241],{"className":14242,"style":261},[137],[91,14244,2249],{"className":14245},[142,212],". Por ejemplo, ",[91,14248,14250,14287],{"className":14249},[94],[91,14251,14253],{"className":14252},[98],[100,14254,14255],{"xmlns":102},[104,14256,14257,14284],{},[107,14258,14259,14261,14263,14265,14267,14269,14271,14273,14275,14277,14279,14281],{},[117,14260,119],{},[330,14262,13938],{"stretchy":9868},[330,14264,332],{},[117,14266,119],{},[330,14268,6524],{},[117,14270,980],{},[330,14272,6524],{},[117,14274,458],{},[330,14276,6524],{},[117,14278,326],{},[330,14280,332],{},[117,14282,14283],{},"24",[121,14285,14286],{"encoding":123},"4! = 4 \\times 3 \\times 2 \\times 1 = 24",[91,14288,14290,14311,14329,14347,14365,14383],{"className":14289,"ariaHidden":129},[128],[91,14291,14293,14296,14299,14302,14305,14308],{"className":14292},[133],[91,14294],{"className":14295,"style":7078},[137],[91,14297,119],{"className":14298},[142],[91,14300,13938],{"className":14301},[231],[91,14303],{"className":14304,"style":421},[420],[91,14306,332],{"className":14307},[425],[91,14309],{"className":14310,"style":421},[420],[91,14312,14314,14317,14320,14323,14326],{"className":14313},[133],[91,14315],{"className":14316,"style":1383},[137],[91,14318,119],{"className":14319},[142],[91,14321],{"className":14322,"style":2275},[420],[91,14324,6524],{"className":14325},[2279],[91,14327],{"className":14328,"style":2275},[420],[91,14330,14332,14335,14338,14341,14344],{"className":14331},[133],[91,14333],{"className":14334,"style":1383},[137],[91,14336,980],{"className":14337},[142],[91,14339],{"className":14340,"style":2275},[420],[91,14342,6524],{"className":14343},[2279],[91,14345],{"className":14346,"style":2275},[420],[91,14348,14350,14353,14356,14359,14362],{"className":14349},[133],[91,14351],{"className":14352,"style":1383},[137],[91,14354,458],{"className":14355},[142],[91,14357],{"className":14358,"style":2275},[420],[91,14360,6524],{"className":14361},[2279],[91,14363],{"className":14364,"style":2275},[420],[91,14366,14368,14371,14374,14377,14380],{"className":14367},[133],[91,14369],{"className":14370,"style":435},[137],[91,14372,326],{"className":14373},[142],[91,14375],{"className":14376,"style":421},[420],[91,14378,332],{"className":14379},[425],[91,14381],{"className":14382,"style":421},[420],[91,14384,14386,14389],{"className":14385},[133],[91,14387],{"className":14388,"style":435},[137],[91,14390,14283],{"className":14391},[142],". Por convención, ",[91,14394,14396,14416],{"className":14395},[94],[91,14397,14399],{"className":14398},[98],[100,14400,14401],{"xmlns":102},[104,14402,14403,14413],{},[107,14404,14405,14407,14409,14411],{},[117,14406,571],{},[330,14408,13938],{"stretchy":9868},[330,14410,332],{},[117,14412,326],{},[121,14414,14415],{"encoding":123},"0! = 1",[91,14417,14419,14440],{"className":14418,"ariaHidden":129},[128],[91,14420,14422,14425,14428,14431,14434,14437],{"className":14421},[133],[91,14423],{"className":14424,"style":7078},[137],[91,14426,571],{"className":14427},[142],[91,14429,13938],{"className":14430},[231],[91,14432],{"className":14433,"style":421},[420],[91,14435,332],{"className":14436},[425],[91,14438],{"className":14439,"style":421},[420],[91,14441,14443,14446],{"className":14442},[133],[91,14444],{"className":14445,"style":435},[137],[91,14447,326],{"className":14448},[142],[17,14450,14451,14452,14891],{},"Comprobemos: ",[91,14453,14455,14527],{"className":14454},[94],[91,14456,14458],{"className":14457},[98],[100,14459,14460],{"xmlns":102},[104,14461,14462,14524],{},[107,14463,14464,14476,14478,14498,14500,14512,14514,14520,14522],{},[107,14465,14466,14468,14474],{},[330,14467,1081],{"fence":129},[110,14469,14470,14472],{"linethickness":4373},[117,14471,119],{},[117,14473,458],{},[330,14475,1090],{"fence":129},[330,14477,332],{},[110,14479,14480,14486],{},[107,14481,14482,14484],{},[117,14483,119],{},[330,14485,13938],{"stretchy":9868},[107,14487,14488,14490,14492,14494,14496],{},[117,14489,458],{},[330,14491,13938],{"stretchy":9868},[330,14493,7396],{},[117,14495,458],{},[330,14497,13938],{"stretchy":9868},[330,14499,332],{},[110,14501,14502,14504],{},[117,14503,14283],{},[107,14505,14506,14508,14510],{},[117,14507,458],{},[330,14509,6524],{},[117,14511,458],{},[330,14513,332],{},[110,14515,14516,14518],{},[117,14517,14283],{},[117,14519,119],{},[330,14521,332],{},[117,14523,5494],{},[121,14525,14526],{"encoding":123},"\\binom{4}{2} = \\frac{4!}{2! \\cdot 2!} = \\frac{24}{2 \\times 2} = \\frac{24}{4} = 6",[91,14528,14530,14611,14710,14799,14882],{"className":14529,"ariaHidden":129},[128],[91,14531,14533,14536,14602,14605,14608],{"className":14532},[133],[91,14534],{"className":14535,"style":4644},[137],[91,14537,14539,14545,14596],{"className":14538},[142],[91,14540,14542],{"className":14541,"style":1126},[146,1125],[91,14543,1081],{"className":14544},[1130,1131],[91,14546,14548],{"className":14547},[110],[91,14549,14551,14588],{"className":14550},[154,155],[91,14552,14554,14585],{"className":14553},[159],[91,14555,14557,14571],{"className":14556,"style":4552},[163],[91,14558,14559,14562],{"style":4555},[91,14560],{"className":14561,"style":1226},[171],[91,14563,14565],{"className":14564},[176,177,178,179],[91,14566,14568],{"className":14567},[142,179],[91,14569,458],{"className":14570},[142,179],[91,14572,14573,14576],{"style":4570},[91,14574],{"className":14575,"style":1226},[171],[91,14577,14579],{"className":14578},[176,177,178,179],[91,14580,14582],{"className":14581},[142,179],[91,14583,119],{"className":14584},[142,179],[91,14586,218],{"className":14587},[217],[91,14589,14591],{"className":14590},[159],[91,14592,14594],{"className":14593,"style":225},[163],[91,14595],{},[91,14597,14599],{"className":14598,"style":1126},[231,1125],[91,14600,1090],{"className":14601},[1130,1131],[91,14603],{"className":14604,"style":421},[420],[91,14606,332],{"className":14607},[425],[91,14609],{"className":14610,"style":421},[420],[91,14612,14614,14618,14701,14704,14707],{"className":14613},[133],[91,14615],{"className":14616,"style":14617},[137],"height:1.2251em;vertical-align:-0.345em;",[91,14619,14621,14624,14698],{"className":14620},[142],[91,14622],{"className":14623},[146,147],[91,14625,14627],{"className":14626},[110],[91,14628,14630,14690],{"className":14629},[154,155],[91,14631,14633,14687],{"className":14632},[159],[91,14634,14636,14662,14670],{"className":14635,"style":14071},[163],[91,14637,14638,14641],{"style":167},[91,14639],{"className":14640,"style":172},[171],[91,14642,14644],{"className":14643},[176,177,178,179],[91,14645,14647,14650,14653,14656,14659],{"className":14646},[142,179],[91,14648,458],{"className":14649},[142,179],[91,14651,13938],{"className":14652},[231,179],[91,14654,7396],{"className":14655},[2279,179],[91,14657,458],{"className":14658},[142,179],[91,14660,13938],{"className":14661},[231,179],[91,14663,14664,14667],{"style":188},[91,14665],{"className":14666,"style":172},[171],[91,14668],{"className":14669,"style":196},[195],[91,14671,14672,14675],{"style":199},[91,14673],{"className":14674,"style":172},[171],[91,14676,14678],{"className":14677},[176,177,178,179],[91,14679,14681,14684],{"className":14680},[142,179],[91,14682,119],{"className":14683},[142,179],[91,14685,13938],{"className":14686},[231,179],[91,14688,218],{"className":14689},[217],[91,14691,14693],{"className":14692},[159],[91,14694,14696],{"className":14695,"style":225},[163],[91,14697],{},[91,14699],{"className":14700},[231,147],[91,14702],{"className":14703,"style":421},[420],[91,14705,332],{"className":14706},[425],[91,14708],{"className":14709,"style":421},[420],[91,14711,14713,14716,14790,14793,14796],{"className":14712},[133],[91,14714],{"className":14715,"style":3709},[137],[91,14717,14719,14722,14787],{"className":14718},[142],[91,14720],{"className":14721},[146,147],[91,14723,14725],{"className":14724},[110],[91,14726,14728,14779],{"className":14727},[154,155],[91,14729,14731,14776],{"className":14730},[159],[91,14732,14734,14754,14762],{"className":14733,"style":366},[163],[91,14735,14736,14739],{"style":167},[91,14737],{"className":14738,"style":172},[171],[91,14740,14742],{"className":14741},[176,177,178,179],[91,14743,14745,14748,14751],{"className":14744},[142,179],[91,14746,458],{"className":14747},[142,179],[91,14749,6524],{"className":14750},[2279,179],[91,14752,458],{"className":14753},[142,179],[91,14755,14756,14759],{"style":188},[91,14757],{"className":14758,"style":172},[171],[91,14760],{"className":14761,"style":196},[195],[91,14763,14764,14767],{"style":199},[91,14765],{"className":14766,"style":172},[171],[91,14768,14770],{"className":14769},[176,177,178,179],[91,14771,14773],{"className":14772},[142,179],[91,14774,14283],{"className":14775},[142,179],[91,14777,218],{"className":14778},[217],[91,14780,14782],{"className":14781},[159],[91,14783,14785],{"className":14784,"style":3788},[163],[91,14786],{},[91,14788],{"className":14789},[231,147],[91,14791],{"className":14792,"style":421},[420],[91,14794,332],{"className":14795},[425],[91,14797],{"className":14798,"style":421},[420],[91,14800,14802,14805,14873,14876,14879],{"className":14801},[133],[91,14803],{"className":14804,"style":347},[137],[91,14806,14808,14811,14870],{"className":14807},[142],[91,14809],{"className":14810},[146,147],[91,14812,14814],{"className":14813},[110],[91,14815,14817,14862],{"className":14816},[154,155],[91,14818,14820,14859],{"className":14819},[159],[91,14821,14823,14837,14845],{"className":14822,"style":366},[163],[91,14824,14825,14828],{"style":167},[91,14826],{"className":14827,"style":172},[171],[91,14829,14831],{"className":14830},[176,177,178,179],[91,14832,14834],{"className":14833},[142,179],[91,14835,119],{"className":14836},[142,179],[91,14838,14839,14842],{"style":188},[91,14840],{"className":14841,"style":172},[171],[91,14843],{"className":14844,"style":196},[195],[91,14846,14847,14850],{"style":199},[91,14848],{"className":14849,"style":172},[171],[91,14851,14853],{"className":14852},[176,177,178,179],[91,14854,14856],{"className":14855},[142,179],[91,14857,14283],{"className":14858},[142,179],[91,14860,218],{"className":14861},[217],[91,14863,14865],{"className":14864},[159],[91,14866,14868],{"className":14867,"style":225},[163],[91,14869],{},[91,14871],{"className":14872},[231,147],[91,14874],{"className":14875,"style":421},[420],[91,14877,332],{"className":14878},[425],[91,14880],{"className":14881,"style":421},[420],[91,14883,14885,14888],{"className":14884},[133],[91,14886],{"className":14887,"style":435},[137],[91,14889,5494],{"className":14890},[142],". Correcto.",[17,14893,14894,14895,14923],{},"Así, la fila ",[91,14896,14898,14911],{"className":14897},[94],[91,14899,14901],{"className":14900},[98],[100,14902,14903],{"xmlns":102},[104,14904,14905,14909],{},[107,14906,14907],{},[113,14908,2249],{},[121,14910,2249],{"encoding":123},[91,14912,14914],{"className":14913,"ariaHidden":129},[128],[91,14915,14917,14920],{"className":14916},[133],[91,14918],{"className":14919,"style":261},[137],[91,14921,2249],{"className":14922},[142,212]," del triángulo contiene los valores:",[17,14925,14927],{"className":14926},[645],[91,14928,14930,15008],{"className":14929},[94],[91,14931,14933],{"className":14932},[98],[100,14934,14935],{"xmlns":102},[104,14936,14937,15005],{},[107,14938,14939,14951,14953,14955,14967,14969,14971,14983,14985,14987,14989,14991,14993],{},[107,14940,14941,14943,14949],{},[330,14942,1081],{"fence":129},[110,14944,14945,14947],{"linethickness":4373},[113,14946,2249],{},[117,14948,571],{},[330,14950,1090],{"fence":129},[330,14952,579],{"separator":129},[659,14954,1450],{},[107,14956,14957,14959,14965],{},[330,14958,1081],{"fence":129},[110,14960,14961,14963],{"linethickness":4373},[113,14962,2249],{},[117,14964,326],{},[330,14966,1090],{"fence":129},[330,14968,579],{"separator":129},[659,14970,1450],{},[107,14972,14973,14975,14981],{},[330,14974,1081],{"fence":129},[110,14976,14977,14979],{"linethickness":4373},[113,14978,2249],{},[117,14980,458],{},[330,14982,1090],{"fence":129},[330,14984,579],{"separator":129},[659,14986,1450],{},[330,14988,4060],{},[330,14990,579],{"separator":129},[659,14992,1450],{},[107,14994,14995,14997,15003],{},[330,14996,1081],{"fence":129},[110,14998,14999,15001],{"linethickness":4373},[113,15000,2249],{},[113,15002,2249],{},[330,15004,1090],{"fence":129},[121,15006,15007],{"encoding":123},"\\binom{n}{0},\\; \\binom{n}{1},\\; \\binom{n}{2},\\; \\ldots,\\; \\binom{n}{n}",[91,15009,15011],{"className":15010,"ariaHidden":129},[128],[91,15012,15014,15017,15083,15086,15089,15092,15158,15161,15164,15167,15233,15236,15239,15242,15245,15248,15251,15254,15257],{"className":15013},[133],[91,15015],{"className":15016,"style":13554},[137],[91,15018,15020,15026,15077],{"className":15019},[142],[91,15021,15023],{"className":15022,"style":1126},[146,1125],[91,15024,1081],{"className":15025},[1130,1131],[91,15027,15029],{"className":15028},[110],[91,15030,15032,15069],{"className":15031},[154,155],[91,15033,15035,15066],{"className":15034},[159],[91,15036,15038,15052],{"className":15037,"style":13576},[163],[91,15039,15040,15043],{"style":4555},[91,15041],{"className":15042,"style":1226},[171],[91,15044,15046],{"className":15045},[176,177,178,179],[91,15047,15049],{"className":15048},[142,179],[91,15050,571],{"className":15051},[142,179],[91,15053,15054,15057],{"style":4570},[91,15055],{"className":15056,"style":1226},[171],[91,15058,15060],{"className":15059},[176,177,178,179],[91,15061,15063],{"className":15062},[142,179],[91,15064,2249],{"className":15065},[142,212,179],[91,15067,218],{"className":15068},[217],[91,15070,15072],{"className":15071},[159],[91,15073,15075],{"className":15074,"style":225},[163],[91,15076],{},[91,15078,15080],{"className":15079,"style":1126},[231,1125],[91,15081,1090],{"className":15082},[1130,1131],[91,15084,579],{"className":15085},[578],[91,15087],{"className":15088,"style":421},[420],[91,15090],{"className":15091,"style":1504},[420],[91,15093,15095,15101,15152],{"className":15094},[142],[91,15096,15098],{"className":15097,"style":1126},[146,1125],[91,15099,1081],{"className":15100},[1130,1131],[91,15102,15104],{"className":15103},[110],[91,15105,15107,15144],{"className":15106},[154,155],[91,15108,15110,15141],{"className":15109},[159],[91,15111,15113,15127],{"className":15112,"style":13576},[163],[91,15114,15115,15118],{"style":4555},[91,15116],{"className":15117,"style":1226},[171],[91,15119,15121],{"className":15120},[176,177,178,179],[91,15122,15124],{"className":15123},[142,179],[91,15125,326],{"className":15126},[142,179],[91,15128,15129,15132],{"style":4570},[91,15130],{"className":15131,"style":1226},[171],[91,15133,15135],{"className":15134},[176,177,178,179],[91,15136,15138],{"className":15137},[142,179],[91,15139,2249],{"className":15140},[142,212,179],[91,15142,218],{"className":15143},[217],[91,15145,15147],{"className":15146},[159],[91,15148,15150],{"className":15149,"style":225},[163],[91,15151],{},[91,15153,15155],{"className":15154,"style":1126},[231,1125],[91,15156,1090],{"className":15157},[1130,1131],[91,15159,579],{"className":15160},[578],[91,15162],{"className":15163,"style":421},[420],[91,15165],{"className":15166,"style":1504},[420],[91,15168,15170,15176,15227],{"className":15169},[142],[91,15171,15173],{"className":15172,"style":1126},[146,1125],[91,15174,1081],{"className":15175},[1130,1131],[91,15177,15179],{"className":15178},[110],[91,15180,15182,15219],{"className":15181},[154,155],[91,15183,15185,15216],{"className":15184},[159],[91,15186,15188,15202],{"className":15187,"style":13576},[163],[91,15189,15190,15193],{"style":4555},[91,15191],{"className":15192,"style":1226},[171],[91,15194,15196],{"className":15195},[176,177,178,179],[91,15197,15199],{"className":15198},[142,179],[91,15200,458],{"className":15201},[142,179],[91,15203,15204,15207],{"style":4570},[91,15205],{"className":15206,"style":1226},[171],[91,15208,15210],{"className":15209},[176,177,178,179],[91,15211,15213],{"className":15212},[142,179],[91,15214,2249],{"className":15215},[142,212,179],[91,15217,218],{"className":15218},[217],[91,15220,15222],{"className":15221},[159],[91,15223,15225],{"className":15224,"style":225},[163],[91,15226],{},[91,15228,15230],{"className":15229,"style":1126},[231,1125],[91,15231,1090],{"className":15232},[1130,1131],[91,15234,579],{"className":15235},[578],[91,15237],{"className":15238,"style":421},[420],[91,15240],{"className":15241,"style":1504},[420],[91,15243,4060],{"className":15244},[1118],[91,15246],{"className":15247,"style":1504},[420],[91,15249,579],{"className":15250},[578],[91,15252],{"className":15253,"style":421},[420],[91,15255],{"className":15256,"style":1504},[420],[91,15258,15260,15266,15317],{"className":15259},[142],[91,15261,15263],{"className":15262,"style":1126},[146,1125],[91,15264,1081],{"className":15265},[1130,1131],[91,15267,15269],{"className":15268},[110],[91,15270,15272,15309],{"className":15271},[154,155],[91,15273,15275,15306],{"className":15274},[159],[91,15276,15278,15292],{"className":15277,"style":13576},[163],[91,15279,15280,15283],{"style":4555},[91,15281],{"className":15282,"style":1226},[171],[91,15284,15286],{"className":15285},[176,177,178,179],[91,15287,15289],{"className":15288},[142,179],[91,15290,2249],{"className":15291},[142,212,179],[91,15293,15294,15297],{"style":4570},[91,15295],{"className":15296,"style":1226},[171],[91,15298,15300],{"className":15299},[176,177,178,179],[91,15301,15303],{"className":15302},[142,179],[91,15304,2249],{"className":15305},[142,212,179],[91,15307,218],{"className":15308},[217],[91,15310,15312],{"className":15311},[159],[91,15313,15315],{"className":15314,"style":225},[163],[91,15316],{},[91,15318,15320],{"className":15319,"style":1126},[231,1125],[91,15321,1090],{"className":15322},[1130,1131],[12,15324,15326],{"id":15325},"la-combinatoria-el-arte-de-contar-sin-enumerar","La combinatoria: el arte de contar sin enumerar",[17,15328,15329,15330,15333,15334,40],{},"La ",[33,15331,15332],{},"combinatoria"," es la rama de las matemáticas que estudia cuántas maneras distintas existen de organizar, agrupar o elegir elementos. Su poder reside en que nos permite conocer el número de posibilidades ",[267,15335,15336],{},"sin tener que listarlas una por una",[17,15338,15339,15340,15343,15344,15347,15348,15376,15377,15450,15451,40],{},"Para entender por qué esto es útil, pensemos en lanzar una moneda. Cada lanzamiento tiene dos resultados posibles: ",[33,15341,15342],{},"cara"," (O) o ",[33,15345,15346],{},"cruz"," (X). Si lanzamos la moneda una vez, hay ",[91,15349,15351,15364],{"className":15350},[94],[91,15352,15354],{"className":15353},[98],[100,15355,15356],{"xmlns":102},[104,15357,15358,15362],{},[107,15359,15360],{},[117,15361,458],{},[121,15363,458],{"encoding":123},[91,15365,15367],{"className":15366,"ariaHidden":129},[128],[91,15368,15370,15373],{"className":15369},[133],[91,15371],{"className":15372,"style":435},[137],[91,15374,458],{"className":15375},[142]," resultados. Si la lanzamos dos veces, cada uno de los 2 resultados del primer lanzamiento se combina con los 2 del segundo, dando ",[91,15378,15380,15402],{"className":15379},[94],[91,15381,15383],{"className":15382},[98],[100,15384,15385],{"xmlns":102},[104,15386,15387,15399],{},[107,15388,15389,15391,15393,15395,15397],{},[117,15390,458],{},[330,15392,6524],{},[117,15394,458],{},[330,15396,332],{},[117,15398,119],{},[121,15400,15401],{"encoding":123},"2 \\times 2 = 4",[91,15403,15405,15423,15441],{"className":15404,"ariaHidden":129},[128],[91,15406,15408,15411,15414,15417,15420],{"className":15407},[133],[91,15409],{"className":15410,"style":1383},[137],[91,15412,458],{"className":15413},[142],[91,15415],{"className":15416,"style":2275},[420],[91,15418,6524],{"className":15419},[2279],[91,15421],{"className":15422,"style":2275},[420],[91,15424,15426,15429,15432,15435,15438],{"className":15425},[133],[91,15427],{"className":15428,"style":435},[137],[91,15430,458],{"className":15431},[142],[91,15433],{"className":15434,"style":421},[420],[91,15436,332],{"className":15437},[425],[91,15439],{"className":15440,"style":421},[420],[91,15442,15444,15447],{"className":15443},[133],[91,15445],{"className":15446,"style":435},[137],[91,15448,119],{"className":15449},[142]," resultados. Con tres lanzamientos, ",[91,15452,15454,15480],{"className":15453},[94],[91,15455,15457],{"className":15456},[98],[100,15458,15459],{"xmlns":102},[104,15460,15461,15477],{},[107,15462,15463,15465,15467,15469,15471,15473,15475],{},[117,15464,458],{},[330,15466,6524],{},[117,15468,458],{},[330,15470,6524],{},[117,15472,458],{},[330,15474,332],{},[117,15476,1101],{},[121,15478,15479],{"encoding":123},"2 \\times 2 \\times 2 = 8",[91,15481,15483,15501,15519,15537],{"className":15482,"ariaHidden":129},[128],[91,15484,15486,15489,15492,15495,15498],{"className":15485},[133],[91,15487],{"className":15488,"style":1383},[137],[91,15490,458],{"className":15491},[142],[91,15493],{"className":15494,"style":2275},[420],[91,15496,6524],{"className":15497},[2279],[91,15499],{"className":15500,"style":2275},[420],[91,15502,15504,15507,15510,15513,15516],{"className":15503},[133],[91,15505],{"className":15506,"style":1383},[137],[91,15508,458],{"className":15509},[142],[91,15511],{"className":15512,"style":2275},[420],[91,15514,6524],{"className":15515},[2279],[91,15517],{"className":15518,"style":2275},[420],[91,15520,15522,15525,15528,15531,15534],{"className":15521},[133],[91,15523],{"className":15524,"style":435},[137],[91,15526,458],{"className":15527},[142],[91,15529],{"className":15530,"style":421},[420],[91,15532,332],{"className":15533},[425],[91,15535],{"className":15536,"style":421},[420],[91,15538,15540,15543],{"className":15539},[133],[91,15541],{"className":15542,"style":435},[137],[91,15544,1101],{"className":15545},[142],[17,15547,15548,15549,11867],{},"Ese patrón se expresa con un ",[33,15550,15551],{},"exponente",[17,15553,15555],{"className":15554},[645],[91,15556,15558,15576],{"className":15557},[94],[91,15559,15561],{"className":15560},[98],[100,15562,15563],{"xmlns":102},[104,15564,15565,15573],{},[107,15566,15567],{},[1075,15568,15569,15571],{},[117,15570,458],{},[113,15572,2249],{},[121,15574,15575],{"encoding":123},"2^n",[91,15577,15579],{"className":15578,"ariaHidden":129},[128],[91,15580,15582,15586],{"className":15581},[133],[91,15583],{"className":15584,"style":15585},[137],"height:0.6644em;",[91,15587,15589,15592],{"className":15588},[142],[91,15590,458],{"className":15591},[142],[91,15593,15595],{"className":15594},[1209],[91,15596,15598],{"className":15597},[154],[91,15599,15601],{"className":15600},[159],[91,15602,15604],{"className":15603,"style":15585},[163],[91,15605,15607,15610],{"style":15606},"top:-3.063em;margin-right:0.05em;",[91,15608],{"className":15609,"style":1226},[171],[91,15611,15613],{"className":15612},[176,177,178,179],[91,15614,2249],{"className":15615},[142,212,179],[17,15617,4605,15618,15646,15647,15675],{},[91,15619,15621,15634],{"className":15620},[94],[91,15622,15624],{"className":15623},[98],[100,15625,15626],{"xmlns":102},[104,15627,15628,15632],{},[107,15629,15630],{},[113,15631,2249],{},[121,15633,2249],{"encoding":123},[91,15635,15637],{"className":15636,"ariaHidden":129},[128],[91,15638,15640,15643],{"className":15639},[133],[91,15641],{"className":15642,"style":261},[137],[91,15644,2249],{"className":15645},[142,212]," es el número de lanzamientos. El exponente indica cuántas veces multiplicamos la base (",[91,15648,15650,15663],{"className":15649},[94],[91,15651,15653],{"className":15652},[98],[100,15654,15655],{"xmlns":102},[104,15656,15657,15661],{},[107,15658,15659],{},[117,15660,458],{},[121,15662,458],{"encoding":123},[91,15664,15666],{"className":15665,"ariaHidden":129},[128],[91,15667,15669,15672],{"className":15668},[133],[91,15670],{"className":15671,"style":435},[137],[91,15673,458],{"className":15674},[142],") por sí misma. Así:",[2220,15677,15678,15774],{},[2223,15679,15680],{},[2226,15681,15682,15713],{},[2229,15683,15684,15685,1090],{"align":2231},"Lanzamientos (",[91,15686,15688,15701],{"className":15687},[94],[91,15689,15691],{"className":15690},[98],[100,15692,15693],{"xmlns":102},[104,15694,15695,15699],{},[107,15696,15697],{},[113,15698,2249],{},[121,15700,2249],{"encoding":123},[91,15702,15704],{"className":15703,"ariaHidden":129},[128],[91,15705,15707,15710],{"className":15706},[133],[91,15708],{"className":15709,"style":261},[137],[91,15711,2249],{"className":15712},[142,212],[2229,15714,15715,15716,1090],{"align":2231},"Resultados posibles (",[91,15717,15719,15736],{"className":15718},[94],[91,15720,15722],{"className":15721},[98],[100,15723,15724],{"xmlns":102},[104,15725,15726,15734],{},[107,15727,15728],{},[1075,15729,15730,15732],{},[117,15731,458],{},[113,15733,2249],{},[121,15735,15575],{"encoding":123},[91,15737,15739],{"className":15738,"ariaHidden":129},[128],[91,15740,15742,15745],{"className":15741},[133],[91,15743],{"className":15744,"style":15585},[137],[91,15746,15748,15751],{"className":15747},[142],[91,15749,458],{"className":15750},[142],[91,15752,15754],{"className":15753},[1209],[91,15755,15757],{"className":15756},[154],[91,15758,15760],{"className":15759},[159],[91,15761,15763],{"className":15762,"style":15585},[163],[91,15764,15765,15768],{"style":15606},[91,15766],{"className":15767,"style":1226},[171],[91,15769,15771],{"className":15770},[176,177,178,179],[91,15772,2249],{"className":15773},[142,212,179],[2302,15775,15776,15782,15788,15794,15800],{},[2226,15777,15778,15780],{},[2307,15779,326],{"align":2231},[2307,15781,458],{"align":2231},[2226,15783,15784,15786],{},[2307,15785,458],{"align":2231},[2307,15787,119],{"align":2231},[2226,15789,15790,15792],{},[2307,15791,980],{"align":2231},[2307,15793,1101],{"align":2231},[2226,15795,15796,15798],{},[2307,15797,583],{"align":2231},[2307,15799,1899],{"align":2231},[2226,15801,15802,15805],{},[2307,15803,15804],{"align":2231},"10",[2307,15806,15807],{"align":2231},"1.024",[17,15809,15810],{},"Con solo 10 lanzamientos ya hay más de mil combinaciones posibles. La combinatoria nos ayuda a manejar esas cantidades sin perder la cabeza.",[17,15812,15813,15814,40],{},"Si necesitas repasar qué es la probabilidad y cómo se mide, echa un vistazo a ",[24,15815,15818],{"href":15816,"rel":15817},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Ffundamentos-de-probabilidad\u002F",[81],"Fundamentos de probabilidad",[12,15820,15822],{"id":15821},"caso-práctico-3-lanzamientos","Caso práctico: 3 lanzamientos",[17,15824,15825,15826,15908],{},"Vamos a conectar todo con un ejemplo concreto. Lanzamos una moneda 3 veces. Sabemos que hay ",[91,15827,15829,15851],{"className":15828},[94],[91,15830,15832],{"className":15831},[98],[100,15833,15834],{"xmlns":102},[104,15835,15836,15848],{},[107,15837,15838,15844,15846],{},[1075,15839,15840,15842],{},[117,15841,458],{},[117,15843,980],{},[330,15845,332],{},[117,15847,1101],{},[121,15849,15850],{"encoding":123},"2^3 = 8",[91,15852,15854,15899],{"className":15853,"ariaHidden":129},[128],[91,15855,15857,15861,15890,15893,15896],{"className":15856},[133],[91,15858],{"className":15859,"style":15860},[137],"height:0.8141em;",[91,15862,15864,15867],{"className":15863},[142],[91,15865,458],{"className":15866},[142],[91,15868,15870],{"className":15869},[1209],[91,15871,15873],{"className":15872},[154],[91,15874,15876],{"className":15875},[159],[91,15877,15879],{"className":15878,"style":15860},[163],[91,15880,15881,15884],{"style":15606},[91,15882],{"className":15883,"style":1226},[171],[91,15885,15887],{"className":15886},[176,177,178,179],[91,15888,980],{"className":15889},[142,179],[91,15891],{"className":15892,"style":421},[420],[91,15894,332],{"className":15895},[425],[91,15897],{"className":15898,"style":421},[420],[91,15900,15902,15905],{"className":15901},[133],[91,15903],{"className":15904,"style":435},[137],[91,15906,1101],{"className":15907},[142]," resultados posibles. Aquí están todos:",[1339,15910,15913],{"className":15911,"code":15912,"language":678,"meta":51},[1342],"OOO    OOX    OXO    XOO    OXX    XOX    XXO    XXX\n",[1345,15914,15912],{"__ignoreMap":51},[17,15916,15917,15918,15921],{},"Ahora agrupemos estos resultados según el ",[33,15919,15920],{},"número de caras"," obtenidas:",[2220,15923,15924,15937],{},[2223,15925,15926],{},[2226,15927,15928,15931,15934],{},[2229,15929,15930],{"align":2231},"Caras",[2229,15932,15933],{"align":7816},"Combinaciones",[2229,15935,15936],{"align":2231},"Cantidad",[2302,15938,15939,15948,15957,15966],{},[2226,15940,15941,15943,15946],{},[2307,15942,571],{"align":2231},[2307,15944,15945],{"align":7816},"XXX",[2307,15947,326],{"align":2231},[2226,15949,15950,15952,15955],{},[2307,15951,326],{"align":2231},[2307,15953,15954],{"align":7816},"OXX, XOX, XXO",[2307,15956,980],{"align":2231},[2226,15958,15959,15961,15964],{},[2307,15960,458],{"align":2231},[2307,15962,15963],{"align":7816},"OOX, OXO, XOO",[2307,15965,980],{"align":2231},[2226,15967,15968,15970,15973],{},[2307,15969,980],{"align":2231},[2307,15971,15972],{"align":7816},"OOO",[2307,15974,326],{"align":2231},[17,15976,15977,15978,16046,16047,16050],{},"Mira la columna de cantidades: ",[91,15979,15981,16007],{"className":15980},[94],[91,15982,15984],{"className":15983},[98],[100,15985,15986],{"xmlns":102},[104,15987,15988,16004],{},[107,15989,15990,15992,15994,15996,15998,16000,16002],{},[117,15991,326],{},[330,15993,579],{"separator":129},[117,15995,980],{},[330,15997,579],{"separator":129},[117,15999,980],{},[330,16001,579],{"separator":129},[117,16003,326],{},[121,16005,16006],{"encoding":123},"1, 3, 3, 1",[91,16008,16010],{"className":16009,"ariaHidden":129},[128],[91,16011,16013,16016,16019,16022,16025,16028,16031,16034,16037,16040,16043],{"className":16012},[133],[91,16014],{"className":16015,"style":567},[137],[91,16017,326],{"className":16018},[142],[91,16020,579],{"className":16021},[578],[91,16023],{"className":16024,"style":1504},[420],[91,16026,980],{"className":16027},[142],[91,16029,579],{"className":16030},[578],[91,16032],{"className":16033,"style":1504},[420],[91,16035,980],{"className":16036},[142],[91,16038,579],{"className":16039},[578],[91,16041],{"className":16042,"style":1504},[420],[91,16044,326],{"className":16045},[142],". ¿Te suena? Es exactamente la ",[33,16048,16049],{},"fila 3"," del triángulo de Tartaglia.",[17,16052,16053,16054,4874,16082,16110,16111,16210,16211,16239,16240,16268],{},"Y no es casualidad. El número de formas de obtener exactamente ",[91,16055,16057,16070],{"className":16056},[94],[91,16058,16060],{"className":16059},[98],[100,16061,16062],{"xmlns":102},[104,16063,16064,16068],{},[107,16065,16066],{},[113,16067,13539],{},[121,16069,13539],{"encoding":123},[91,16071,16073],{"className":16072,"ariaHidden":129},[128],[91,16074,16076,16079],{"className":16075},[133],[91,16077],{"className":16078,"style":7078},[137],[91,16080,13539],{"className":16081,"style":13591},[142,212],[91,16083,16085,16098],{"className":16084},[94],[91,16086,16088],{"className":16087},[98],[100,16089,16090],{"xmlns":102},[104,16091,16092,16096],{},[107,16093,16094],{},[113,16095,2249],{},[121,16097,2249],{"encoding":123},[91,16099,16101],{"className":16100,"ariaHidden":129},[128],[91,16102,16104,16107],{"className":16103},[133],[91,16105],{"className":16106,"style":261},[137],[91,16108,2249],{"className":16109},[142,212]," lanzamientos es precisamente ",[91,16112,16114,16135],{"className":16113},[94],[91,16115,16117],{"className":16116},[98],[100,16118,16119],{"xmlns":102},[104,16120,16121,16133],{},[107,16122,16123,16125,16131],{},[330,16124,1081],{"fence":129},[110,16126,16127,16129],{"linethickness":4373},[113,16128,2249],{},[113,16130,13539],{},[330,16132,1090],{"fence":129},[121,16134,13544],{"encoding":123},[91,16136,16138],{"className":16137,"ariaHidden":129},[128],[91,16139,16141,16144],{"className":16140},[133],[91,16142],{"className":16143,"style":13554},[137],[91,16145,16147,16153,16204],{"className":16146},[142],[91,16148,16150],{"className":16149,"style":1126},[146,1125],[91,16151,1081],{"className":16152},[1130,1131],[91,16154,16156],{"className":16155},[110],[91,16157,16159,16196],{"className":16158},[154,155],[91,16160,16162,16193],{"className":16161},[159],[91,16163,16165,16179],{"className":16164,"style":13576},[163],[91,16166,16167,16170],{"style":4555},[91,16168],{"className":16169,"style":1226},[171],[91,16171,16173],{"className":16172},[176,177,178,179],[91,16174,16176],{"className":16175},[142,179],[91,16177,13539],{"className":16178,"style":13591},[142,212,179],[91,16180,16181,16184],{"style":4570},[91,16182],{"className":16183,"style":1226},[171],[91,16185,16187],{"className":16186},[176,177,178,179],[91,16188,16190],{"className":16189},[142,179],[91,16191,2249],{"className":16192},[142,212,179],[91,16194,218],{"className":16195},[217],[91,16197,16199],{"className":16198},[159],[91,16200,16202],{"className":16201,"style":225},[163],[91,16203],{},[91,16205,16207],{"className":16206,"style":1126},[231,1125],[91,16208,1090],{"className":16209},[1130,1131]," — el coeficiente binomial que ocupa la posición ",[91,16212,16214,16227],{"className":16213},[94],[91,16215,16217],{"className":16216},[98],[100,16218,16219],{"xmlns":102},[104,16220,16221,16225],{},[107,16222,16223],{},[113,16224,13539],{},[121,16226,13539],{"encoding":123},[91,16228,16230],{"className":16229,"ariaHidden":129},[128],[91,16231,16233,16236],{"className":16232},[133],[91,16234],{"className":16235,"style":7078},[137],[91,16237,13539],{"className":16238,"style":13591},[142,212]," en la fila ",[91,16241,16243,16256],{"className":16242},[94],[91,16244,16246],{"className":16245},[98],[100,16247,16248],{"xmlns":102},[104,16249,16250,16254],{},[107,16251,16252],{},[113,16253,2249],{},[121,16255,2249],{"encoding":123},[91,16257,16259],{"className":16258,"ariaHidden":129},[128],[91,16260,16262,16265],{"className":16261},[133],[91,16263],{"className":16264,"style":261},[137],[91,16266,2249],{"className":16267},[142,212]," del triángulo.",[635,16270,16272],{"id":16271},"la-probabilidad-entra-en-escena","La probabilidad entra en escena",[17,16274,15329,16275,16277],{},[33,16276,6521],{}," de un suceso mide qué tan posible es que ocurra. Se calcula como:",[17,16279,16281],{"className":16280},[645],[91,16282,16284,16309],{"className":16283},[94],[91,16285,16287],{"className":16286},[98],[100,16288,16289],{"xmlns":102},[104,16290,16291,16306],{},[107,16292,16293,16296,16298],{},[113,16294,16295],{},"P",[330,16297,332],{},[110,16299,16300,16303],{},[659,16301,16302],{},"casos favorables",[659,16304,16305],{},"casos posibles",[121,16307,16308],{"encoding":123},"P = \\frac{\\text{casos favorables}}{\\text{casos posibles}}",[91,16310,16312,16331],{"className":16311,"ariaHidden":129},[128],[91,16313,16315,16318,16322,16325,16328],{"className":16314},[133],[91,16316],{"className":16317,"style":6878},[137],[91,16319,16295],{"className":16320,"style":16321},[142,212],"margin-right:0.1389em;",[91,16323],{"className":16324,"style":421},[420],[91,16326,332],{"className":16327},[425],[91,16329],{"className":16330,"style":421},[420],[91,16332,16334,16338],{"className":16333},[133],[91,16335],{"className":16336,"style":16337},[137],"height:1.3612em;vertical-align:-0.4811em;",[91,16339,16341,16344,16410],{"className":16340},[142],[91,16342],{"className":16343},[146,147],[91,16345,16347],{"className":16346},[110],[91,16348,16350,16401],{"className":16349},[154,155],[91,16351,16353,16398],{"className":16352},[159],[91,16354,16356,16373,16381],{"className":16355,"style":14071},[163],[91,16357,16358,16361],{"style":167},[91,16359],{"className":16360,"style":172},[171],[91,16362,16364],{"className":16363},[176,177,178,179],[91,16365,16367],{"className":16366},[142,179],[91,16368,16370],{"className":16369},[142,678,179],[91,16371,16305],{"className":16372},[142,179],[91,16374,16375,16378],{"style":188},[91,16376],{"className":16377,"style":172},[171],[91,16379],{"className":16380,"style":196},[195],[91,16382,16383,16386],{"style":199},[91,16384],{"className":16385,"style":172},[171],[91,16387,16389],{"className":16388},[176,177,178,179],[91,16390,16392],{"className":16391},[142,179],[91,16393,16395],{"className":16394},[142,678,179],[91,16396,16302],{"className":16397},[142,179],[91,16399,218],{"className":16400},[217],[91,16402,16404],{"className":16403},[159],[91,16405,16408],{"className":16406,"style":16407},[163],"height:0.4811em;",[91,16409],{},[91,16411],{"className":16412},[231,147],[17,16414,16415,16416,16444],{},"Si la moneda es justa, cada uno de los 8 resultados es igual de probable. Así que la probabilidad de obtener exactamente ",[91,16417,16419,16432],{"className":16418},[94],[91,16420,16422],{"className":16421},[98],[100,16423,16424],{"xmlns":102},[104,16425,16426,16430],{},[107,16427,16428],{},[113,16429,13539],{},[121,16431,13539],{"encoding":123},[91,16433,16435],{"className":16434,"ariaHidden":129},[128],[91,16436,16438,16441],{"className":16437},[133],[91,16439],{"className":16440,"style":7078},[137],[91,16442,13539],{"className":16443,"style":13591},[142,212]," caras en 3 lanzamientos es:",[17,16446,16448],{"className":16447},[645],[91,16449,16451,16514],{"className":16450},[94],[91,16452,16454],{"className":16453},[98],[100,16455,16456],{"xmlns":102},[104,16457,16458,16511],{},[107,16459,16460,16462,16464,16466,16469,16471,16473,16493,16495],{},[113,16461,16295],{},[330,16463,1081],{"stretchy":9868},[113,16465,13539],{},[659,16467,16468],{}," caras",[330,16470,1090],{"stretchy":9868},[330,16472,332],{},[110,16474,16475,16487],{},[107,16476,16477,16479,16485],{},[330,16478,1081],{"fence":129},[110,16480,16481,16483],{"linethickness":4373},[117,16482,980],{},[113,16484,13539],{},[330,16486,1090],{"fence":129},[1075,16488,16489,16491],{},[117,16490,458],{},[117,16492,980],{},[330,16494,332],{},[110,16496,16497,16509],{},[107,16498,16499,16501,16507],{},[330,16500,1081],{"fence":129},[110,16502,16503,16505],{"linethickness":4373},[117,16504,980],{},[113,16506,13539],{},[330,16508,1090],{"fence":129},[117,16510,1101],{},[121,16512,16513],{"encoding":123},"P(k \\text{ caras}) = \\frac{\\binom{3}{k}}{2^3} = \\frac{\\binom{3}{k}}{8}",[91,16515,16517,16550,16728],{"className":16516,"ariaHidden":129},[128],[91,16518,16520,16523,16526,16529,16532,16538,16541,16544,16547],{"className":16519},[133],[91,16521],{"className":16522,"style":9889},[137],[91,16524,16295],{"className":16525,"style":16321},[142,212],[91,16527,1081],{"className":16528},[146],[91,16530,13539],{"className":16531,"style":13591},[142,212],[91,16533,16535],{"className":16534},[142,678],[91,16536,16468],{"className":16537},[142],[91,16539,1090],{"className":16540},[231],[91,16542],{"className":16543,"style":421},[420],[91,16545,332],{"className":16546},[425],[91,16548],{"className":16549,"style":421},[420],[91,16551,16553,16556,16719,16722,16725],{"className":16552},[133],[91,16554],{"className":16555,"style":11357},[137],[91,16557,16559,16562,16716],{"className":16558},[142],[91,16560],{"className":16561},[146,147],[91,16563,16565],{"className":16564},[110],[91,16566,16568,16708],{"className":16567},[154,155],[91,16569,16571,16705],{"className":16570},[159],[91,16572,16574,16614,16622],{"className":16573,"style":10300},[163],[91,16575,16576,16579],{"style":167},[91,16577],{"className":16578,"style":172},[171],[91,16580,16582],{"className":16581},[176,177,178,179],[91,16583,16585],{"className":16584},[142,179],[91,16586,16588,16591],{"className":16587},[142,179],[91,16589,458],{"className":16590},[142,179],[91,16592,16594],{"className":16593},[1209],[91,16595,16597],{"className":16596},[154],[91,16598,16600],{"className":16599},[159],[91,16601,16603],{"className":16602,"style":7573},[163],[91,16604,16605,16608],{"style":7576},[91,16606],{"className":16607,"style":7580},[171],[91,16609,16611],{"className":16610},[176,7584,1131,179],[91,16612,980],{"className":16613},[142,179],[91,16615,16616,16619],{"style":188},[91,16617],{"className":16618,"style":172},[171],[91,16620],{"className":16621,"style":196},[195],[91,16623,16624,16627],{"style":10334},[91,16625],{"className":16626,"style":172},[171],[91,16628,16630],{"className":16629},[176,177,178,179],[91,16631,16633],{"className":16632},[142,179],[91,16634,16636,16645,16696],{"className":16635},[142,179],[91,16637,16639],{"className":16638,"style":10351},[146,176,7584,10350,179,1125],[91,16640,16642],{"className":16641},[1130,1131,179],[91,16643,1081],{"className":16644},[179],[91,16646,16648],{"className":16647},[110],[91,16649,16651,16688],{"className":16650},[154,155],[91,16652,16654,16685],{"className":16653},[159],[91,16655,16657,16671],{"className":16656,"style":10370},[163],[91,16658,16659,16662],{"style":10373},[91,16660],{"className":16661,"style":7580},[171],[91,16663,16665],{"className":16664},[176,7584,1131,179],[91,16666,16668],{"className":16667},[142,179],[91,16669,13539],{"className":16670,"style":13591},[142,212,179],[91,16672,16673,16676],{"style":10388},[91,16674],{"className":16675,"style":7580},[171],[91,16677,16679],{"className":16678},[176,7584,1131,179],[91,16680,16682],{"className":16681},[142,179],[91,16683,980],{"className":16684},[142,179],[91,16686,218],{"className":16687},[217],[91,16689,16691],{"className":16690},[159],[91,16692,16694],{"className":16693,"style":10413},[163],[91,16695],{},[91,16697,16699],{"className":16698,"style":10351},[231,176,7584,10350,179,1125],[91,16700,16702],{"className":16701},[1130,1131,179],[91,16703,1090],{"className":16704},[179],[91,16706,218],{"className":16707},[217],[91,16709,16711],{"className":16710},[159],[91,16712,16714],{"className":16713,"style":225},[163],[91,16715],{},[91,16717],{"className":16718},[231,147],[91,16720],{"className":16721,"style":421},[420],[91,16723,332],{"className":16724},[425],[91,16726],{"className":16727,"style":421},[420],[91,16729,16731,16734],{"className":16730},[133],[91,16732],{"className":16733,"style":11357},[137],[91,16735,16737,16740,16868],{"className":16736},[142],[91,16738],{"className":16739},[146,147],[91,16741,16743],{"className":16742},[110],[91,16744,16746,16860],{"className":16745},[154,155],[91,16747,16749,16857],{"className":16748},[159],[91,16750,16752,16766,16774],{"className":16751,"style":10300},[163],[91,16753,16754,16757],{"style":167},[91,16755],{"className":16756,"style":172},[171],[91,16758,16760],{"className":16759},[176,177,178,179],[91,16761,16763],{"className":16762},[142,179],[91,16764,1101],{"className":16765},[142,179],[91,16767,16768,16771],{"style":188},[91,16769],{"className":16770,"style":172},[171],[91,16772],{"className":16773,"style":196},[195],[91,16775,16776,16779],{"style":10334},[91,16777],{"className":16778,"style":172},[171],[91,16780,16782],{"className":16781},[176,177,178,179],[91,16783,16785],{"className":16784},[142,179],[91,16786,16788,16797,16848],{"className":16787},[142,179],[91,16789,16791],{"className":16790,"style":10351},[146,176,7584,10350,179,1125],[91,16792,16794],{"className":16793},[1130,1131,179],[91,16795,1081],{"className":16796},[179],[91,16798,16800],{"className":16799},[110],[91,16801,16803,16840],{"className":16802},[154,155],[91,16804,16806,16837],{"className":16805},[159],[91,16807,16809,16823],{"className":16808,"style":10370},[163],[91,16810,16811,16814],{"style":10373},[91,16812],{"className":16813,"style":7580},[171],[91,16815,16817],{"className":16816},[176,7584,1131,179],[91,16818,16820],{"className":16819},[142,179],[91,16821,13539],{"className":16822,"style":13591},[142,212,179],[91,16824,16825,16828],{"style":10388},[91,16826],{"className":16827,"style":7580},[171],[91,16829,16831],{"className":16830},[176,7584,1131,179],[91,16832,16834],{"className":16833},[142,179],[91,16835,980],{"className":16836},[142,179],[91,16838,218],{"className":16839},[217],[91,16841,16843],{"className":16842},[159],[91,16844,16846],{"className":16845,"style":10413},[163],[91,16847],{},[91,16849,16851],{"className":16850,"style":10351},[231,176,7584,10350,179,1125],[91,16852,16854],{"className":16853},[1130,1131,179],[91,16855,1090],{"className":16856},[179],[91,16858,218],{"className":16859},[217],[91,16861,16863],{"className":16862},[159],[91,16864,16866],{"className":16865,"style":225},[163],[91,16867],{},[91,16869],{"className":16870},[231,147],[17,16872,16873],{},"Calculemos cada caso:",[2220,16875,16876,17019],{},[2223,16877,16878],{},[2226,16879,16880,16911,17013,17016],{},[2229,16881,16882,16910],{"align":2231},[91,16883,16885,16898],{"className":16884},[94],[91,16886,16888],{"className":16887},[98],[100,16889,16890],{"xmlns":102},[104,16891,16892,16896],{},[107,16893,16894],{},[113,16895,13539],{},[121,16897,13539],{"encoding":123},[91,16899,16901],{"className":16900,"ariaHidden":129},[128],[91,16902,16904,16907],{"className":16903},[133],[91,16905],{"className":16906,"style":7078},[137],[91,16908,13539],{"className":16909,"style":13591},[142,212]," caras",[2229,16912,16913],{"align":2231},[91,16914,16916,16938],{"className":16915},[94],[91,16917,16919],{"className":16918},[98],[100,16920,16921],{"xmlns":102},[104,16922,16923,16935],{},[107,16924,16925,16927,16933],{},[330,16926,1081],{"fence":129},[110,16928,16929,16931],{"linethickness":4373},[117,16930,980],{},[113,16932,13539],{},[330,16934,1090],{"fence":129},[121,16936,16937],{"encoding":123},"\\binom{3}{k}",[91,16939,16941],{"className":16940,"ariaHidden":129},[128],[91,16942,16944,16947],{"className":16943},[133],[91,16945],{"className":16946,"style":4644},[137],[91,16948,16950,16956,17007],{"className":16949},[142],[91,16951,16953],{"className":16952,"style":1126},[146,1125],[91,16954,1081],{"className":16955},[1130,1131],[91,16957,16959],{"className":16958},[110],[91,16960,16962,16999],{"className":16961},[154,155],[91,16963,16965,16996],{"className":16964},[159],[91,16966,16968,16982],{"className":16967,"style":4552},[163],[91,16969,16970,16973],{"style":4555},[91,16971],{"className":16972,"style":1226},[171],[91,16974,16976],{"className":16975},[176,177,178,179],[91,16977,16979],{"className":16978},[142,179],[91,16980,13539],{"className":16981,"style":13591},[142,212,179],[91,16983,16984,16987],{"style":4570},[91,16985],{"className":16986,"style":1226},[171],[91,16988,16990],{"className":16989},[176,177,178,179],[91,16991,16993],{"className":16992},[142,179],[91,16994,980],{"className":16995},[142,179],[91,16997,218],{"className":16998},[217],[91,17000,17002],{"className":17001},[159],[91,17003,17005],{"className":17004,"style":225},[163],[91,17006],{},[91,17008,17010],{"className":17009,"style":1126},[231,1125],[91,17011,1090],{"className":17012},[1130,1131],[2229,17014,17015],{"align":2231},"Probabilidad",[2229,17017,17018],{"align":2231},"Porcentaje",[2302,17020,17021,17163,17305,17443],{},[2226,17022,17023,17025,17027,17160],{},[2307,17024,571],{"align":2231},[2307,17026,326],{"align":2231},[2307,17028,17029],{"align":2231},[91,17030,17032,17055],{"className":17031},[94],[91,17033,17035],{"className":17034},[98],[100,17036,17037],{"xmlns":102},[104,17038,17039,17052],{},[107,17040,17041,17047,17049],{},[110,17042,17043,17045],{},[117,17044,326],{},[117,17046,1101],{},[330,17048,332],{},[117,17050,17051],{},"0,125",[121,17053,17054],{"encoding":123},"\\frac{1}{8} = 0{,}125",[91,17056,17058,17141],{"className":17057,"ariaHidden":129},[128],[91,17059,17061,17064,17132,17135,17138],{"className":17060},[133],[91,17062],{"className":17063,"style":347},[137],[91,17065,17067,17070,17129],{"className":17066},[142],[91,17068],{"className":17069},[146,147],[91,17071,17073],{"className":17072},[110],[91,17074,17076,17121],{"className":17075},[154,155],[91,17077,17079,17118],{"className":17078},[159],[91,17080,17082,17096,17104],{"className":17081,"style":366},[163],[91,17083,17084,17087],{"style":167},[91,17085],{"className":17086,"style":172},[171],[91,17088,17090],{"className":17089},[176,177,178,179],[91,17091,17093],{"className":17092},[142,179],[91,17094,1101],{"className":17095},[142,179],[91,17097,17098,17101],{"style":188},[91,17099],{"className":17100,"style":172},[171],[91,17102],{"className":17103,"style":196},[195],[91,17105,17106,17109],{"style":199},[91,17107],{"className":17108,"style":172},[171],[91,17110,17112],{"className":17111},[176,177,178,179],[91,17113,17115],{"className":17114},[142,179],[91,17116,326],{"className":17117},[142,179],[91,17119,218],{"className":17120},[217],[91,17122,17124],{"className":17123},[159],[91,17125,17127],{"className":17126,"style":225},[163],[91,17128],{},[91,17130],{"className":17131},[231,147],[91,17133],{"className":17134,"style":421},[420],[91,17136,332],{"className":17137},[425],[91,17139],{"className":17140,"style":421},[420],[91,17142,17144,17147,17150,17156],{"className":17143},[133],[91,17145],{"className":17146,"style":567},[137],[91,17148,571],{"className":17149},[142],[91,17151,17153],{"className":17152},[142],[91,17154,579],{"className":17155},[578],[91,17157,17159],{"className":17158},[142],"125",[2307,17161,17162],{"align":2231},"12,5 %",[2226,17164,17165,17167,17169,17302],{},[2307,17166,326],{"align":2231},[2307,17168,980],{"align":2231},[2307,17170,17171],{"align":2231},[91,17172,17174,17197],{"className":17173},[94],[91,17175,17177],{"className":17176},[98],[100,17178,17179],{"xmlns":102},[104,17180,17181,17194],{},[107,17182,17183,17189,17191],{},[110,17184,17185,17187],{},[117,17186,980],{},[117,17188,1101],{},[330,17190,332],{},[117,17192,17193],{},"0,375",[121,17195,17196],{"encoding":123},"\\frac{3}{8} = 0{,}375",[91,17198,17200,17283],{"className":17199,"ariaHidden":129},[128],[91,17201,17203,17206,17274,17277,17280],{"className":17202},[133],[91,17204],{"className":17205,"style":347},[137],[91,17207,17209,17212,17271],{"className":17208},[142],[91,17210],{"className":17211},[146,147],[91,17213,17215],{"className":17214},[110],[91,17216,17218,17263],{"className":17217},[154,155],[91,17219,17221,17260],{"className":17220},[159],[91,17222,17224,17238,17246],{"className":17223,"style":366},[163],[91,17225,17226,17229],{"style":167},[91,17227],{"className":17228,"style":172},[171],[91,17230,17232],{"className":17231},[176,177,178,179],[91,17233,17235],{"className":17234},[142,179],[91,17236,1101],{"className":17237},[142,179],[91,17239,17240,17243],{"style":188},[91,17241],{"className":17242,"style":172},[171],[91,17244],{"className":17245,"style":196},[195],[91,17247,17248,17251],{"style":199},[91,17249],{"className":17250,"style":172},[171],[91,17252,17254],{"className":17253},[176,177,178,179],[91,17255,17257],{"className":17256},[142,179],[91,17258,980],{"className":17259},[142,179],[91,17261,218],{"className":17262},[217],[91,17264,17266],{"className":17265},[159],[91,17267,17269],{"className":17268,"style":225},[163],[91,17270],{},[91,17272],{"className":17273},[231,147],[91,17275],{"className":17276,"style":421},[420],[91,17278,332],{"className":17279},[425],[91,17281],{"className":17282,"style":421},[420],[91,17284,17286,17289,17292,17298],{"className":17285},[133],[91,17287],{"className":17288,"style":567},[137],[91,17290,571],{"className":17291},[142],[91,17293,17295],{"className":17294},[142],[91,17296,579],{"className":17297},[578],[91,17299,17301],{"className":17300},[142],"375",[2307,17303,17304],{"align":2231},"37,5 %",[2226,17306,17307,17309,17311,17441],{},[2307,17308,458],{"align":2231},[2307,17310,980],{"align":2231},[2307,17312,17313],{"align":2231},[91,17314,17316,17337],{"className":17315},[94],[91,17317,17319],{"className":17318},[98],[100,17320,17321],{"xmlns":102},[104,17322,17323,17335],{},[107,17324,17325,17331,17333],{},[110,17326,17327,17329],{},[117,17328,980],{},[117,17330,1101],{},[330,17332,332],{},[117,17334,17193],{},[121,17336,17196],{"encoding":123},[91,17338,17340,17423],{"className":17339,"ariaHidden":129},[128],[91,17341,17343,17346,17414,17417,17420],{"className":17342},[133],[91,17344],{"className":17345,"style":347},[137],[91,17347,17349,17352,17411],{"className":17348},[142],[91,17350],{"className":17351},[146,147],[91,17353,17355],{"className":17354},[110],[91,17356,17358,17403],{"className":17357},[154,155],[91,17359,17361,17400],{"className":17360},[159],[91,17362,17364,17378,17386],{"className":17363,"style":366},[163],[91,17365,17366,17369],{"style":167},[91,17367],{"className":17368,"style":172},[171],[91,17370,17372],{"className":17371},[176,177,178,179],[91,17373,17375],{"className":17374},[142,179],[91,17376,1101],{"className":17377},[142,179],[91,17379,17380,17383],{"style":188},[91,17381],{"className":17382,"style":172},[171],[91,17384],{"className":17385,"style":196},[195],[91,17387,17388,17391],{"style":199},[91,17389],{"className":17390,"style":172},[171],[91,17392,17394],{"className":17393},[176,177,178,179],[91,17395,17397],{"className":17396},[142,179],[91,17398,980],{"className":17399},[142,179],[91,17401,218],{"className":17402},[217],[91,17404,17406],{"className":17405},[159],[91,17407,17409],{"className":17408,"style":225},[163],[91,17410],{},[91,17412],{"className":17413},[231,147],[91,17415],{"className":17416,"style":421},[420],[91,17418,332],{"className":17419},[425],[91,17421],{"className":17422,"style":421},[420],[91,17424,17426,17429,17432,17438],{"className":17425},[133],[91,17427],{"className":17428,"style":567},[137],[91,17430,571],{"className":17431},[142],[91,17433,17435],{"className":17434},[142],[91,17436,579],{"className":17437},[578],[91,17439,17301],{"className":17440},[142],[2307,17442,17304],{"align":2231},[2226,17444,17445,17447,17449,17579],{},[2307,17446,980],{"align":2231},[2307,17448,326],{"align":2231},[2307,17450,17451],{"align":2231},[91,17452,17454,17475],{"className":17453},[94],[91,17455,17457],{"className":17456},[98],[100,17458,17459],{"xmlns":102},[104,17460,17461,17473],{},[107,17462,17463,17469,17471],{},[110,17464,17465,17467],{},[117,17466,326],{},[117,17468,1101],{},[330,17470,332],{},[117,17472,17051],{},[121,17474,17054],{"encoding":123},[91,17476,17478,17561],{"className":17477,"ariaHidden":129},[128],[91,17479,17481,17484,17552,17555,17558],{"className":17480},[133],[91,17482],{"className":17483,"style":347},[137],[91,17485,17487,17490,17549],{"className":17486},[142],[91,17488],{"className":17489},[146,147],[91,17491,17493],{"className":17492},[110],[91,17494,17496,17541],{"className":17495},[154,155],[91,17497,17499,17538],{"className":17498},[159],[91,17500,17502,17516,17524],{"className":17501,"style":366},[163],[91,17503,17504,17507],{"style":167},[91,17505],{"className":17506,"style":172},[171],[91,17508,17510],{"className":17509},[176,177,178,179],[91,17511,17513],{"className":17512},[142,179],[91,17514,1101],{"className":17515},[142,179],[91,17517,17518,17521],{"style":188},[91,17519],{"className":17520,"style":172},[171],[91,17522],{"className":17523,"style":196},[195],[91,17525,17526,17529],{"style":199},[91,17527],{"className":17528,"style":172},[171],[91,17530,17532],{"className":17531},[176,177,178,179],[91,17533,17535],{"className":17534},[142,179],[91,17536,326],{"className":17537},[142,179],[91,17539,218],{"className":17540},[217],[91,17542,17544],{"className":17543},[159],[91,17545,17547],{"className":17546,"style":225},[163],[91,17548],{},[91,17550],{"className":17551},[231,147],[91,17553],{"className":17554,"style":421},[420],[91,17556,332],{"className":17557},[425],[91,17559],{"className":17560,"style":421},[420],[91,17562,17564,17567,17570,17576],{"className":17563},[133],[91,17565],{"className":17566,"style":567},[137],[91,17568,571],{"className":17569},[142],[91,17571,17573],{"className":17572},[142],[91,17574,579],{"className":17575},[578],[91,17577,17159],{"className":17578},[142],[2307,17580,17162],{"align":2231},[17,17582,17583,17584,17737,17738,17766],{},"Si sumamos todas las probabilidades: ",[91,17585,17587,17617],{"className":17586},[94],[91,17588,17590],{"className":17589},[98],[100,17591,17592],{"xmlns":102},[104,17593,17594,17614],{},[107,17595,17596,17598,17600,17602,17604,17606,17608,17610,17612],{},[117,17597,17051],{},[330,17599,1368],{},[117,17601,17193],{},[330,17603,1368],{},[117,17605,17193],{},[330,17607,1368],{},[117,17609,17051],{},[330,17611,332],{},[117,17613,326],{},[121,17615,17616],{"encoding":123},"0{,}125 + 0{,}375 + 0{,}375 + 0{,}125 = 1",[91,17618,17620,17647,17674,17701,17728],{"className":17619,"ariaHidden":129},[128],[91,17621,17623,17626,17629,17635,17638,17641,17644],{"className":17622},[133],[91,17624],{"className":17625,"style":567},[137],[91,17627,571],{"className":17628},[142],[91,17630,17632],{"className":17631},[142],[91,17633,579],{"className":17634},[578],[91,17636,17159],{"className":17637},[142],[91,17639],{"className":17640,"style":2275},[420],[91,17642,1368],{"className":17643},[2279],[91,17645],{"className":17646,"style":2275},[420],[91,17648,17650,17653,17656,17662,17665,17668,17671],{"className":17649},[133],[91,17651],{"className":17652,"style":567},[137],[91,17654,571],{"className":17655},[142],[91,17657,17659],{"className":17658},[142],[91,17660,579],{"className":17661},[578],[91,17663,17301],{"className":17664},[142],[91,17666],{"className":17667,"style":2275},[420],[91,17669,1368],{"className":17670},[2279],[91,17672],{"className":17673,"style":2275},[420],[91,17675,17677,17680,17683,17689,17692,17695,17698],{"className":17676},[133],[91,17678],{"className":17679,"style":567},[137],[91,17681,571],{"className":17682},[142],[91,17684,17686],{"className":17685},[142],[91,17687,579],{"className":17688},[578],[91,17690,17301],{"className":17691},[142],[91,17693],{"className":17694,"style":2275},[420],[91,17696,1368],{"className":17697},[2279],[91,17699],{"className":17700,"style":2275},[420],[91,17702,17704,17707,17710,17716,17719,17722,17725],{"className":17703},[133],[91,17705],{"className":17706,"style":567},[137],[91,17708,571],{"className":17709},[142],[91,17711,17713],{"className":17712},[142],[91,17714,579],{"className":17715},[578],[91,17717,17159],{"className":17718},[142],[91,17720],{"className":17721,"style":421},[420],[91,17723,332],{"className":17724},[425],[91,17726],{"className":17727,"style":421},[420],[91,17729,17731,17734],{"className":17730},[133],[91,17732],{"className":17733,"style":435},[137],[91,17735,326],{"className":17736},[142],". La suma siempre da ",[91,17739,17741,17754],{"className":17740},[94],[91,17742,17744],{"className":17743},[98],[100,17745,17746],{"xmlns":102},[104,17747,17748,17752],{},[107,17749,17750],{},[117,17751,326],{},[121,17753,326],{"encoding":123},[91,17755,17757],{"className":17756,"ariaHidden":129},[128],[91,17758,17760,17763],{"className":17759},[133],[91,17761],{"className":17762,"style":435},[137],[91,17764,326],{"className":17765},[142],", porque entre todos los casos se cubre el 100 % de las posibilidades.",[17,17768,17769],{},"La fórmula general que acabamos de descubrir funciona para cualquier número de lanzamientos:",[17,17771,17773],{"className":17772},[645],[91,17774,17776,17826],{"className":17775},[94],[91,17777,17779],{"className":17778},[98],[100,17780,17781],{"xmlns":102},[104,17782,17783,17823],{},[107,17784,17785,17787,17789,17791,17794,17796,17799,17801,17803],{},[113,17786,16295],{},[330,17788,1081],{"stretchy":9868},[113,17790,13539],{},[659,17792,17793],{}," caras en ",[113,17795,2249],{},[659,17797,17798],{}," lanzamientos",[330,17800,1090],{"stretchy":9868},[330,17802,332],{},[110,17804,17805,17817],{},[107,17806,17807,17809,17815],{},[330,17808,1081],{"fence":129},[110,17810,17811,17813],{"linethickness":4373},[113,17812,2249],{},[113,17814,13539],{},[330,17816,1090],{"fence":129},[1075,17818,17819,17821],{},[117,17820,458],{},[113,17822,2249],{},[121,17824,17825],{"encoding":123},"P(k \\text{ caras en } n \\text{ lanzamientos}) = \\frac{\\binom{n}{k}}{2^n}",[91,17827,17829,17871],{"className":17828,"ariaHidden":129},[128],[91,17830,17832,17835,17838,17841,17844,17850,17853,17859,17862,17865,17868],{"className":17831},[133],[91,17833],{"className":17834,"style":9889},[137],[91,17836,16295],{"className":17837,"style":16321},[142,212],[91,17839,1081],{"className":17840},[146],[91,17842,13539],{"className":17843,"style":13591},[142,212],[91,17845,17847],{"className":17846},[142,678],[91,17848,17793],{"className":17849},[142],[91,17851,2249],{"className":17852},[142,212],[91,17854,17856],{"className":17855},[142,678],[91,17857,17798],{"className":17858},[142],[91,17860,1090],{"className":17861},[231],[91,17863],{"className":17864,"style":421},[420],[91,17866,332],{"className":17867},[425],[91,17869],{"className":17870,"style":421},[420],[91,17872,17874,17877],{"className":17873},[133],[91,17875],{"className":17876,"style":11357},[137],[91,17878,17880,17883,18038],{"className":17879},[142],[91,17881],{"className":17882},[146,147],[91,17884,17886],{"className":17885},[110],[91,17887,17889,18030],{"className":17888},[154,155],[91,17890,17892,18027],{"className":17891},[159],[91,17893,17895,17935,17943],{"className":17894,"style":10300},[163],[91,17896,17897,17900],{"style":167},[91,17898],{"className":17899,"style":172},[171],[91,17901,17903],{"className":17902},[176,177,178,179],[91,17904,17906],{"className":17905},[142,179],[91,17907,17909,17912],{"className":17908},[142,179],[91,17910,458],{"className":17911},[142,179],[91,17913,17915],{"className":17914},[1209],[91,17916,17918],{"className":17917},[154],[91,17919,17921],{"className":17920},[159],[91,17922,17924],{"className":17923,"style":11077},[163],[91,17925,17926,17929],{"style":7576},[91,17927],{"className":17928,"style":7580},[171],[91,17930,17932],{"className":17931},[176,7584,1131,179],[91,17933,2249],{"className":17934},[142,212,179],[91,17936,17937,17940],{"style":188},[91,17938],{"className":17939,"style":172},[171],[91,17941],{"className":17942,"style":196},[195],[91,17944,17945,17948],{"style":10334},[91,17946],{"className":17947,"style":172},[171],[91,17949,17951],{"className":17950},[176,177,178,179],[91,17952,17954],{"className":17953},[142,179],[91,17955,17957,17966,18018],{"className":17956},[142,179],[91,17958,17960],{"className":17959,"style":10351},[146,176,7584,10350,179,1125],[91,17961,17963],{"className":17962},[1130,1131,179],[91,17964,1081],{"className":17965},[179],[91,17967,17969],{"className":17968},[110],[91,17970,17972,18010],{"className":17971},[154,155],[91,17973,17975,18007],{"className":17974},[159],[91,17976,17979,17993],{"className":17977,"style":17978},[163],"height:0.7785em;",[91,17980,17981,17984],{"style":10373},[91,17982],{"className":17983,"style":7580},[171],[91,17985,17987],{"className":17986},[176,7584,1131,179],[91,17988,17990],{"className":17989},[142,179],[91,17991,13539],{"className":17992,"style":13591},[142,212,179],[91,17994,17995,17998],{"style":10388},[91,17996],{"className":17997,"style":7580},[171],[91,17999,18001],{"className":18000},[176,7584,1131,179],[91,18002,18004],{"className":18003},[142,179],[91,18005,2249],{"className":18006},[142,212,179],[91,18008,218],{"className":18009},[217],[91,18011,18013],{"className":18012},[159],[91,18014,18016],{"className":18015,"style":10413},[163],[91,18017],{},[91,18019,18021],{"className":18020,"style":10351},[231,176,7584,10350,179,1125],[91,18022,18024],{"className":18023},[1130,1131,179],[91,18025,1090],{"className":18026},[179],[91,18028,218],{"className":18029},[217],[91,18031,18033],{"className":18032},[159],[91,18034,18036],{"className":18035,"style":225},[163],[91,18037],{},[91,18039],{"className":18040},[231,147],[12,18042,18044],{"id":18043},"caso-práctico-5-lanzamientos","Caso práctico: 5 lanzamientos",[17,18046,18047],{},"Subamos la apuesta. Con 5 lanzamientos, la fila correspondiente del triángulo es:",[17,18049,18051],{"className":18050},[645],[91,18052,18054,18089],{"className":18053},[94],[91,18055,18057],{"className":18056},[98],[100,18058,18059],{"xmlns":102},[104,18060,18061,18086],{},[107,18062,18063,18065,18068,18070,18072,18074,18076,18078,18080,18082,18084],{},[117,18064,326],{},[420,18066],{"width":18067},"1em",[117,18069,583],{},[420,18071],{"width":18067},[117,18073,15804],{},[420,18075],{"width":18067},[117,18077,15804],{},[420,18079],{"width":18067},[117,18081,583],{},[420,18083],{"width":18067},[117,18085,326],{},[121,18087,18088],{"encoding":123},"1 \\quad 5 \\quad 10 \\quad 10 \\quad 5 \\quad 1",[91,18090,18092],{"className":18091,"ariaHidden":129},[128],[91,18093,18095,18098,18101,18105,18108,18111,18114,18117,18120,18123,18126,18129],{"className":18094},[133],[91,18096],{"className":18097,"style":435},[137],[91,18099,326],{"className":18100},[142],[91,18102],{"className":18103,"style":18104},[420],"margin-right:1em;",[91,18106,583],{"className":18107},[142],[91,18109],{"className":18110,"style":18104},[420],[91,18112,15804],{"className":18113},[142],[91,18115],{"className":18116,"style":18104},[420],[91,18118,15804],{"className":18119},[142],[91,18121],{"className":18122,"style":18104},[420],[91,18124,583],{"className":18125},[142],[91,18127],{"className":18128,"style":18104},[420],[91,18130,326],{"className":18131},[142],[17,18133,18134,18135,40],{},"y el total de resultados posibles es ",[91,18136,18138,18160],{"className":18137},[94],[91,18139,18141],{"className":18140},[98],[100,18142,18143],{"xmlns":102},[104,18144,18145,18157],{},[107,18146,18147,18153,18155],{},[1075,18148,18149,18151],{},[117,18150,458],{},[117,18152,583],{},[330,18154,332],{},[117,18156,1899],{},[121,18158,18159],{"encoding":123},"2^5 = 32",[91,18161,18163,18207],{"className":18162,"ariaHidden":129},[128],[91,18164,18166,18169,18198,18201,18204],{"className":18165},[133],[91,18167],{"className":18168,"style":15860},[137],[91,18170,18172,18175],{"className":18171},[142],[91,18173,458],{"className":18174},[142],[91,18176,18178],{"className":18177},[1209],[91,18179,18181],{"className":18180},[154],[91,18182,18184],{"className":18183},[159],[91,18185,18187],{"className":18186,"style":15860},[163],[91,18188,18189,18192],{"style":15606},[91,18190],{"className":18191,"style":1226},[171],[91,18193,18195],{"className":18194},[176,177,178,179],[91,18196,583],{"className":18197},[142,179],[91,18199],{"className":18200,"style":421},[420],[91,18202,332],{"className":18203},[425],[91,18205],{"className":18206,"style":421},[420],[91,18208,18210,18213],{"className":18209},[133],[91,18211],{"className":18212,"style":435},[137],[91,18214,1899],{"className":18215},[142],[17,18217,18218,18219,18222,18223,18347],{},"Veamos algunos ejemplos concretos. ¿De cuántas formas podemos obtener exactamente ",[33,18220,18221],{},"2 caras","? El triángulo nos dice ",[91,18224,18226,18254],{"className":18225},[94],[91,18227,18229],{"className":18228},[98],[100,18230,18231],{"xmlns":102},[104,18232,18233,18251],{},[107,18234,18235,18247,18249],{},[107,18236,18237,18239,18245],{},[330,18238,1081],{"fence":129},[110,18240,18241,18243],{"linethickness":4373},[117,18242,583],{},[117,18244,458],{},[330,18246,1090],{"fence":129},[330,18248,332],{},[117,18250,15804],{},[121,18252,18253],{"encoding":123},"\\binom{5}{2} = 10",[91,18255,18257,18338],{"className":18256,"ariaHidden":129},[128],[91,18258,18260,18263,18329,18332,18335],{"className":18259},[133],[91,18261],{"className":18262,"style":4644},[137],[91,18264,18266,18272,18323],{"className":18265},[142],[91,18267,18269],{"className":18268,"style":1126},[146,1125],[91,18270,1081],{"className":18271},[1130,1131],[91,18273,18275],{"className":18274},[110],[91,18276,18278,18315],{"className":18277},[154,155],[91,18279,18281,18312],{"className":18280},[159],[91,18282,18284,18298],{"className":18283,"style":4552},[163],[91,18285,18286,18289],{"style":4555},[91,18287],{"className":18288,"style":1226},[171],[91,18290,18292],{"className":18291},[176,177,178,179],[91,18293,18295],{"className":18294},[142,179],[91,18296,458],{"className":18297},[142,179],[91,18299,18300,18303],{"style":4570},[91,18301],{"className":18302,"style":1226},[171],[91,18304,18306],{"className":18305},[176,177,178,179],[91,18307,18309],{"className":18308},[142,179],[91,18310,583],{"className":18311},[142,179],[91,18313,218],{"className":18314},[217],[91,18316,18318],{"className":18317},[159],[91,18319,18321],{"className":18320,"style":225},[163],[91,18322],{},[91,18324,18326],{"className":18325,"style":1126},[231,1125],[91,18327,1090],{"className":18328},[1130,1131],[91,18330],{"className":18331,"style":421},[420],[91,18333,332],{"className":18334},[425],[91,18336],{"className":18337,"style":421},[420],[91,18339,18341,18344],{"className":18340},[133],[91,18342],{"className":18343,"style":435},[137],[91,18345,15804],{"className":18346},[142],". Estas son las 10 combinaciones:",[1339,18349,18352],{"className":18350,"code":18351,"language":678,"meta":51},[1342],"OOXXX   OXOXX   OXXOX   OXXXO   XOOXX\nXOXOX   XOXXO   XXOOX   XXOXO   XXXOO\n",[1345,18353,18351],{"__ignoreMap":51},[17,18355,18356,18357,18360,18361,18485],{},"¿Y exactamente ",[33,18358,18359],{},"3 caras","? También ",[91,18362,18364,18392],{"className":18363},[94],[91,18365,18367],{"className":18366},[98],[100,18368,18369],{"xmlns":102},[104,18370,18371,18389],{},[107,18372,18373,18385,18387],{},[107,18374,18375,18377,18383],{},[330,18376,1081],{"fence":129},[110,18378,18379,18381],{"linethickness":4373},[117,18380,583],{},[117,18382,980],{},[330,18384,1090],{"fence":129},[330,18386,332],{},[117,18388,15804],{},[121,18390,18391],{"encoding":123},"\\binom{5}{3} = 10",[91,18393,18395,18476],{"className":18394,"ariaHidden":129},[128],[91,18396,18398,18401,18467,18470,18473],{"className":18397},[133],[91,18399],{"className":18400,"style":4644},[137],[91,18402,18404,18410,18461],{"className":18403},[142],[91,18405,18407],{"className":18406,"style":1126},[146,1125],[91,18408,1081],{"className":18409},[1130,1131],[91,18411,18413],{"className":18412},[110],[91,18414,18416,18453],{"className":18415},[154,155],[91,18417,18419,18450],{"className":18418},[159],[91,18420,18422,18436],{"className":18421,"style":4552},[163],[91,18423,18424,18427],{"style":4555},[91,18425],{"className":18426,"style":1226},[171],[91,18428,18430],{"className":18429},[176,177,178,179],[91,18431,18433],{"className":18432},[142,179],[91,18434,980],{"className":18435},[142,179],[91,18437,18438,18441],{"style":4570},[91,18439],{"className":18440,"style":1226},[171],[91,18442,18444],{"className":18443},[176,177,178,179],[91,18445,18447],{"className":18446},[142,179],[91,18448,583],{"className":18449},[142,179],[91,18451,218],{"className":18452},[217],[91,18454,18456],{"className":18455},[159],[91,18457,18459],{"className":18458,"style":225},[163],[91,18460],{},[91,18462,18464],{"className":18463,"style":1126},[231,1125],[91,18465,1090],{"className":18466},[1130,1131],[91,18468],{"className":18469,"style":421},[420],[91,18471,332],{"className":18472},[425],[91,18474],{"className":18475,"style":421},[420],[91,18477,18479,18482],{"className":18478},[133],[91,18480],{"className":18481,"style":435},[137],[91,18483,15804],{"className":18484},[142],". El triángulo es simétrico: hay tantas formas de obtener 2 caras (y 3 cruces) como 3 caras (y 2 cruces).",[17,18487,18488],{},"La tabla completa de probabilidades:",[2220,18490,18491,18631],{},[2223,18492,18493],{},[2226,18494,18495,18525,18627,18629],{},[2229,18496,18497,16910],{"align":2231},[91,18498,18500,18513],{"className":18499},[94],[91,18501,18503],{"className":18502},[98],[100,18504,18505],{"xmlns":102},[104,18506,18507,18511],{},[107,18508,18509],{},[113,18510,13539],{},[121,18512,13539],{"encoding":123},[91,18514,18516],{"className":18515,"ariaHidden":129},[128],[91,18517,18519,18522],{"className":18518},[133],[91,18520],{"className":18521,"style":7078},[137],[91,18523,13539],{"className":18524,"style":13591},[142,212],[2229,18526,18527],{"align":2231},[91,18528,18530,18552],{"className":18529},[94],[91,18531,18533],{"className":18532},[98],[100,18534,18535],{"xmlns":102},[104,18536,18537,18549],{},[107,18538,18539,18541,18547],{},[330,18540,1081],{"fence":129},[110,18542,18543,18545],{"linethickness":4373},[117,18544,583],{},[113,18546,13539],{},[330,18548,1090],{"fence":129},[121,18550,18551],{"encoding":123},"\\binom{5}{k}",[91,18553,18555],{"className":18554,"ariaHidden":129},[128],[91,18556,18558,18561],{"className":18557},[133],[91,18559],{"className":18560,"style":4644},[137],[91,18562,18564,18570,18621],{"className":18563},[142],[91,18565,18567],{"className":18566,"style":1126},[146,1125],[91,18568,1081],{"className":18569},[1130,1131],[91,18571,18573],{"className":18572},[110],[91,18574,18576,18613],{"className":18575},[154,155],[91,18577,18579,18610],{"className":18578},[159],[91,18580,18582,18596],{"className":18581,"style":4552},[163],[91,18583,18584,18587],{"style":4555},[91,18585],{"className":18586,"style":1226},[171],[91,18588,18590],{"className":18589},[176,177,178,179],[91,18591,18593],{"className":18592},[142,179],[91,18594,13539],{"className":18595,"style":13591},[142,212,179],[91,18597,18598,18601],{"style":4570},[91,18599],{"className":18600,"style":1226},[171],[91,18602,18604],{"className":18603},[176,177,178,179],[91,18605,18607],{"className":18606},[142,179],[91,18608,583],{"className":18609},[142,179],[91,18611,218],{"className":18612},[217],[91,18614,18616],{"className":18615},[159],[91,18617,18619],{"className":18618,"style":225},[163],[91,18620],{},[91,18622,18624],{"className":18623,"style":1126},[231,1125],[91,18625,1090],{"className":18626},[1130,1131],[2229,18628,17015],{"align":2231},[2229,18630,17018],{"align":2231},[2302,18632,18633,18741,18850,18959,19066,19173],{},[2226,18634,18635,18637,18639,18738],{},[2307,18636,571],{"align":2231},[2307,18638,326],{"align":2231},[2307,18640,18641],{"align":2231},[91,18642,18644,18661],{"className":18643},[94],[91,18645,18647],{"className":18646},[98],[100,18648,18649],{"xmlns":102},[104,18650,18651,18659],{},[107,18652,18653],{},[110,18654,18655,18657],{},[117,18656,326],{},[117,18658,1899],{},[121,18660,2998],{"encoding":123},[91,18662,18664],{"className":18663,"ariaHidden":129},[128],[91,18665,18667,18670],{"className":18666},[133],[91,18668],{"className":18669,"style":347},[137],[91,18671,18673,18676,18735],{"className":18672},[142],[91,18674],{"className":18675},[146,147],[91,18677,18679],{"className":18678},[110],[91,18680,18682,18727],{"className":18681},[154,155],[91,18683,18685,18724],{"className":18684},[159],[91,18686,18688,18702,18710],{"className":18687,"style":366},[163],[91,18689,18690,18693],{"style":167},[91,18691],{"className":18692,"style":172},[171],[91,18694,18696],{"className":18695},[176,177,178,179],[91,18697,18699],{"className":18698},[142,179],[91,18700,1899],{"className":18701},[142,179],[91,18703,18704,18707],{"style":188},[91,18705],{"className":18706,"style":172},[171],[91,18708],{"className":18709,"style":196},[195],[91,18711,18712,18715],{"style":199},[91,18713],{"className":18714,"style":172},[171],[91,18716,18718],{"className":18717},[176,177,178,179],[91,18719,18721],{"className":18720},[142,179],[91,18722,326],{"className":18723},[142,179],[91,18725,218],{"className":18726},[217],[91,18728,18730],{"className":18729},[159],[91,18731,18733],{"className":18732,"style":225},[163],[91,18734],{},[91,18736],{"className":18737},[231,147],[2307,18739,18740],{"align":2231},"3,1 %",[2226,18742,18743,18745,18747,18847],{},[2307,18744,326],{"align":2231},[2307,18746,583],{"align":2231},[2307,18748,18749],{"align":2231},[91,18750,18752,18770],{"className":18751},[94],[91,18753,18755],{"className":18754},[98],[100,18756,18757],{"xmlns":102},[104,18758,18759,18767],{},[107,18760,18761],{},[110,18762,18763,18765],{},[117,18764,583],{},[117,18766,1899],{},[121,18768,18769],{"encoding":123},"\\frac{5}{32}",[91,18771,18773],{"className":18772,"ariaHidden":129},[128],[91,18774,18776,18779],{"className":18775},[133],[91,18777],{"className":18778,"style":347},[137],[91,18780,18782,18785,18844],{"className":18781},[142],[91,18783],{"className":18784},[146,147],[91,18786,18788],{"className":18787},[110],[91,18789,18791,18836],{"className":18790},[154,155],[91,18792,18794,18833],{"className":18793},[159],[91,18795,18797,18811,18819],{"className":18796,"style":366},[163],[91,18798,18799,18802],{"style":167},[91,18800],{"className":18801,"style":172},[171],[91,18803,18805],{"className":18804},[176,177,178,179],[91,18806,18808],{"className":18807},[142,179],[91,18809,1899],{"className":18810},[142,179],[91,18812,18813,18816],{"style":188},[91,18814],{"className":18815,"style":172},[171],[91,18817],{"className":18818,"style":196},[195],[91,18820,18821,18824],{"style":199},[91,18822],{"className":18823,"style":172},[171],[91,18825,18827],{"className":18826},[176,177,178,179],[91,18828,18830],{"className":18829},[142,179],[91,18831,583],{"className":18832},[142,179],[91,18834,218],{"className":18835},[217],[91,18837,18839],{"className":18838},[159],[91,18840,18842],{"className":18841,"style":225},[163],[91,18843],{},[91,18845],{"className":18846},[231,147],[2307,18848,18849],{"align":2231},"15,6 %",[2226,18851,18852,18854,18856,18956],{},[2307,18853,458],{"align":2231},[2307,18855,15804],{"align":2231},[2307,18857,18858],{"align":2231},[91,18859,18861,18879],{"className":18860},[94],[91,18862,18864],{"className":18863},[98],[100,18865,18866],{"xmlns":102},[104,18867,18868,18876],{},[107,18869,18870],{},[110,18871,18872,18874],{},[117,18873,15804],{},[117,18875,1899],{},[121,18877,18878],{"encoding":123},"\\frac{10}{32}",[91,18880,18882],{"className":18881,"ariaHidden":129},[128],[91,18883,18885,18888],{"className":18884},[133],[91,18886],{"className":18887,"style":347},[137],[91,18889,18891,18894,18953],{"className":18890},[142],[91,18892],{"className":18893},[146,147],[91,18895,18897],{"className":18896},[110],[91,18898,18900,18945],{"className":18899},[154,155],[91,18901,18903,18942],{"className":18902},[159],[91,18904,18906,18920,18928],{"className":18905,"style":366},[163],[91,18907,18908,18911],{"style":167},[91,18909],{"className":18910,"style":172},[171],[91,18912,18914],{"className":18913},[176,177,178,179],[91,18915,18917],{"className":18916},[142,179],[91,18918,1899],{"className":18919},[142,179],[91,18921,18922,18925],{"style":188},[91,18923],{"className":18924,"style":172},[171],[91,18926],{"className":18927,"style":196},[195],[91,18929,18930,18933],{"style":199},[91,18931],{"className":18932,"style":172},[171],[91,18934,18936],{"className":18935},[176,177,178,179],[91,18937,18939],{"className":18938},[142,179],[91,18940,15804],{"className":18941},[142,179],[91,18943,218],{"className":18944},[217],[91,18946,18948],{"className":18947},[159],[91,18949,18951],{"className":18950,"style":225},[163],[91,18952],{},[91,18954],{"className":18955},[231,147],[2307,18957,18958],{"align":2231},"31,3 %",[2226,18960,18961,18963,18965,19064],{},[2307,18962,980],{"align":2231},[2307,18964,15804],{"align":2231},[2307,18966,18967],{"align":2231},[91,18968,18970,18987],{"className":18969},[94],[91,18971,18973],{"className":18972},[98],[100,18974,18975],{"xmlns":102},[104,18976,18977,18985],{},[107,18978,18979],{},[110,18980,18981,18983],{},[117,18982,15804],{},[117,18984,1899],{},[121,18986,18878],{"encoding":123},[91,18988,18990],{"className":18989,"ariaHidden":129},[128],[91,18991,18993,18996],{"className":18992},[133],[91,18994],{"className":18995,"style":347},[137],[91,18997,18999,19002,19061],{"className":18998},[142],[91,19000],{"className":19001},[146,147],[91,19003,19005],{"className":19004},[110],[91,19006,19008,19053],{"className":19007},[154,155],[91,19009,19011,19050],{"className":19010},[159],[91,19012,19014,19028,19036],{"className":19013,"style":366},[163],[91,19015,19016,19019],{"style":167},[91,19017],{"className":19018,"style":172},[171],[91,19020,19022],{"className":19021},[176,177,178,179],[91,19023,19025],{"className":19024},[142,179],[91,19026,1899],{"className":19027},[142,179],[91,19029,19030,19033],{"style":188},[91,19031],{"className":19032,"style":172},[171],[91,19034],{"className":19035,"style":196},[195],[91,19037,19038,19041],{"style":199},[91,19039],{"className":19040,"style":172},[171],[91,19042,19044],{"className":19043},[176,177,178,179],[91,19045,19047],{"className":19046},[142,179],[91,19048,15804],{"className":19049},[142,179],[91,19051,218],{"className":19052},[217],[91,19054,19056],{"className":19055},[159],[91,19057,19059],{"className":19058,"style":225},[163],[91,19060],{},[91,19062],{"className":19063},[231,147],[2307,19065,18958],{"align":2231},[2226,19067,19068,19070,19072,19171],{},[2307,19069,119],{"align":2231},[2307,19071,583],{"align":2231},[2307,19073,19074],{"align":2231},[91,19075,19077,19094],{"className":19076},[94],[91,19078,19080],{"className":19079},[98],[100,19081,19082],{"xmlns":102},[104,19083,19084,19092],{},[107,19085,19086],{},[110,19087,19088,19090],{},[117,19089,583],{},[117,19091,1899],{},[121,19093,18769],{"encoding":123},[91,19095,19097],{"className":19096,"ariaHidden":129},[128],[91,19098,19100,19103],{"className":19099},[133],[91,19101],{"className":19102,"style":347},[137],[91,19104,19106,19109,19168],{"className":19105},[142],[91,19107],{"className":19108},[146,147],[91,19110,19112],{"className":19111},[110],[91,19113,19115,19160],{"className":19114},[154,155],[91,19116,19118,19157],{"className":19117},[159],[91,19119,19121,19135,19143],{"className":19120,"style":366},[163],[91,19122,19123,19126],{"style":167},[91,19124],{"className":19125,"style":172},[171],[91,19127,19129],{"className":19128},[176,177,178,179],[91,19130,19132],{"className":19131},[142,179],[91,19133,1899],{"className":19134},[142,179],[91,19136,19137,19140],{"style":188},[91,19138],{"className":19139,"style":172},[171],[91,19141],{"className":19142,"style":196},[195],[91,19144,19145,19148],{"style":199},[91,19146],{"className":19147,"style":172},[171],[91,19149,19151],{"className":19150},[176,177,178,179],[91,19152,19154],{"className":19153},[142,179],[91,19155,583],{"className":19156},[142,179],[91,19158,218],{"className":19159},[217],[91,19161,19163],{"className":19162},[159],[91,19164,19166],{"className":19165,"style":225},[163],[91,19167],{},[91,19169],{"className":19170},[231,147],[2307,19172,18849],{"align":2231},[2226,19174,19175,19177,19179,19278],{},[2307,19176,583],{"align":2231},[2307,19178,326],{"align":2231},[2307,19180,19181],{"align":2231},[91,19182,19184,19201],{"className":19183},[94],[91,19185,19187],{"className":19186},[98],[100,19188,19189],{"xmlns":102},[104,19190,19191,19199],{},[107,19192,19193],{},[110,19194,19195,19197],{},[117,19196,326],{},[117,19198,1899],{},[121,19200,2998],{"encoding":123},[91,19202,19204],{"className":19203,"ariaHidden":129},[128],[91,19205,19207,19210],{"className":19206},[133],[91,19208],{"className":19209,"style":347},[137],[91,19211,19213,19216,19275],{"className":19212},[142],[91,19214],{"className":19215},[146,147],[91,19217,19219],{"className":19218},[110],[91,19220,19222,19267],{"className":19221},[154,155],[91,19223,19225,19264],{"className":19224},[159],[91,19226,19228,19242,19250],{"className":19227,"style":366},[163],[91,19229,19230,19233],{"style":167},[91,19231],{"className":19232,"style":172},[171],[91,19234,19236],{"className":19235},[176,177,178,179],[91,19237,19239],{"className":19238},[142,179],[91,19240,1899],{"className":19241},[142,179],[91,19243,19244,19247],{"style":188},[91,19245],{"className":19246,"style":172},[171],[91,19248],{"className":19249,"style":196},[195],[91,19251,19252,19255],{"style":199},[91,19253],{"className":19254,"style":172},[171],[91,19256,19258],{"className":19257},[176,177,178,179],[91,19259,19261],{"className":19260},[142,179],[91,19262,326],{"className":19263},[142,179],[91,19265,218],{"className":19266},[217],[91,19268,19270],{"className":19269},[159],[91,19271,19273],{"className":19272,"style":225},[163],[91,19274],{},[91,19276],{"className":19277},[231,147],[2307,19279,18740],{"align":2231},[17,19281,19282],{},"Si lo dibujamos como un histograma, la distribución tiene una forma familiar:",[1339,19284,19287],{"className":19285,"code":19286,"language":678,"meta":51},[1342],"0 caras  ▌              3,1 %\n1 cara   ▌▌▌▌▌         15,6 %\n2 caras  ▌▌▌▌▌▌▌▌▌▌    31,3 %\n3 caras  ▌▌▌▌▌▌▌▌▌▌    31,3 %\n4 caras  ▌▌▌▌▌         15,6 %\n5 caras  ▌              3,1 %\n",[1345,19288,19286],{"__ignoreMap":51},[17,19290,19291],{},"Observa tres cosas:",[284,19293,19294,19300,19306],{},[287,19295,19296,19299],{},[33,19297,19298],{},"Simetría."," La distribución es un espejo perfecto alrededor del centro. Obtener 0 caras es tan improbable como obtener 5; obtener 1 es tan probable como obtener 4.",[287,19301,19302,19305],{},[33,19303,19304],{},"Concentración central."," Los resultados más probables están en el centro (2 o 3 caras), mientras que los extremos (todo caras o todo cruces) son muy raros.",[287,19307,19308,19311],{},[33,19309,19310],{},"Forma de campana."," La silueta recuerda a la famosa curva de Gauss. Esto no es accidental: a medida que aumentamos el número de lanzamientos, la distribución se aproxima cada vez más a esa curva.",[17,19313,19314,19315,40],{},"Si te interesa explorar más la estadística detrás de los lanzamientos, lee ",[24,19316,19319],{"href":19317,"rel":19318},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fla-estadistica-inesperada-de-los-lanzamientos-de-moneda\u002F",[81],"La estadística inesperada de los lanzamientos de moneda",[12,19321,19323],{"id":19322},"la-conexión-con-el-álgebra-el-binomio-de-newton","La conexión con el álgebra: el binomio de Newton",[17,19325,19326,19327,19330],{},"Hasta ahora hemos hablado de triángulos y monedas. Pero hay un tercer protagonista que conecta todo: el ",[33,19328,19329],{},"álgebra",". Concretamente, una expresión que quizá recuerdes del colegio:",[17,19332,19334],{"className":19333},[645],[91,19335,19337,19364],{"className":19336},[94],[91,19338,19340],{"className":19339},[98],[100,19341,19342],{"xmlns":102},[104,19343,19344,19361],{},[107,19345,19346,19348,19350,19352,19355],{},[330,19347,1081],{"stretchy":9868},[113,19349,24],{},[330,19351,1368],{},[113,19353,19354],{},"b",[1075,19356,19357,19359],{},[330,19358,1090],{"stretchy":9868},[113,19360,2249],{},[121,19362,19363],{"encoding":123},"(a + b)^n",[91,19365,19367,19388],{"className":19366,"ariaHidden":129},[128],[91,19368,19370,19373,19376,19379,19382,19385],{"className":19369},[133],[91,19371],{"className":19372,"style":9889},[137],[91,19374,1081],{"className":19375},[146],[91,19377,24],{"className":19378},[142,212],[91,19380],{"className":19381,"style":2275},[420],[91,19383,1368],{"className":19384},[2279],[91,19386],{"className":19387,"style":2275},[420],[91,19389,19391,19394,19397],{"className":19390},[133],[91,19392],{"className":19393,"style":9889},[137],[91,19395,19354],{"className":19396},[142,212],[91,19398,19400,19403],{"className":19399},[231],[91,19401,1090],{"className":19402},[231],[91,19404,19406],{"className":19405},[1209],[91,19407,19409],{"className":19408},[154],[91,19410,19412],{"className":19411},[159],[91,19413,19415],{"className":19414,"style":15585},[163],[91,19416,19417,19420],{"style":15606},[91,19418],{"className":19419,"style":1226},[171],[91,19421,19423],{"className":19422},[176,177,178,179],[91,19424,2249],{"className":19425},[142,212,179],[17,19427,19428],{},"Antes de desarrollarla, aclaremos un par de conceptos:",[3601,19430,19431,19552],{},[287,19432,19433,19434,19437,19438,40],{},"Un ",[33,19435,19436],{},"polinomio"," es una expresión matemática formada por sumas de términos con variables elevadas a distintas potencias. Por ejemplo: ",[91,19439,19441,19471],{"className":19440},[94],[91,19442,19444],{"className":19443},[98],[100,19445,19446],{"xmlns":102},[104,19447,19448,19468],{},[107,19449,19450,19452,19458,19460,19462,19464,19466],{},[117,19451,980],{},[1075,19453,19454,19456],{},[113,19455,11247],{},[117,19457,458],{},[330,19459,1368],{},[117,19461,458],{},[113,19463,11247],{},[330,19465,1368],{},[117,19467,326],{},[121,19469,19470],{"encoding":123},"3x^2 + 2x + 1",[91,19472,19474,19522,19543],{"className":19473,"ariaHidden":129},[128],[91,19475,19477,19481,19484,19513,19516,19519],{"className":19476},[133],[91,19478],{"className":19479,"style":19480},[137],"height:0.8974em;vertical-align:-0.0833em;",[91,19482,980],{"className":19483},[142],[91,19485,19487,19490],{"className":19486},[142],[91,19488,11247],{"className":19489},[142,212],[91,19491,19493],{"className":19492},[1209],[91,19494,19496],{"className":19495},[154],[91,19497,19499],{"className":19498},[159],[91,19500,19502],{"className":19501,"style":15860},[163],[91,19503,19504,19507],{"style":15606},[91,19505],{"className":19506,"style":1226},[171],[91,19508,19510],{"className":19509},[176,177,178,179],[91,19511,458],{"className":19512},[142,179],[91,19514],{"className":19515,"style":2275},[420],[91,19517,1368],{"className":19518},[2279],[91,19520],{"className":19521,"style":2275},[420],[91,19523,19525,19528,19531,19534,19537,19540],{"className":19524},[133],[91,19526],{"className":19527,"style":1383},[137],[91,19529,458],{"className":19530},[142],[91,19532,11247],{"className":19533},[142,212],[91,19535],{"className":19536,"style":2275},[420],[91,19538,1368],{"className":19539},[2279],[91,19541],{"className":19542,"style":2275},[420],[91,19544,19546,19549],{"className":19545},[133],[91,19547],{"className":19548,"style":435},[137],[91,19550,326],{"className":19551},[142],[287,19553,19433,19554,19557,19558,19561,19562,40],{},[33,19555,19556],{},"binomio"," es un caso particular de polinomio: tiene exactamente ",[33,19559,19560],{},"dos"," términos. Por ejemplo: ",[91,19563,19565,19587],{"className":19564},[94],[91,19566,19568],{"className":19567},[98],[100,19569,19570],{"xmlns":102},[104,19571,19572,19584],{},[107,19573,19574,19576,19578,19580,19582],{},[330,19575,1081],{"stretchy":9868},[113,19577,24],{},[330,19579,1368],{},[113,19581,19354],{},[330,19583,1090],{"stretchy":9868},[121,19585,19586],{"encoding":123},"(a + b)",[91,19588,19590,19611],{"className":19589,"ariaHidden":129},[128],[91,19591,19593,19596,19599,19602,19605,19608],{"className":19592},[133],[91,19594],{"className":19595,"style":9889},[137],[91,19597,1081],{"className":19598},[146],[91,19600,24],{"className":19601},[142,212],[91,19603],{"className":19604,"style":2275},[420],[91,19606,1368],{"className":19607},[2279],[91,19609],{"className":19610,"style":2275},[420],[91,19612,19614,19617,19620],{"className":19613},[133],[91,19615],{"className":19616,"style":9889},[137],[91,19618,19354],{"className":19619},[142,212],[91,19621,1090],{"className":19622},[231],[17,19624,19625,19626,19654],{},"Elevar un binomio a una potencia significa multiplicarlo por sí mismo ",[91,19627,19629,19642],{"className":19628},[94],[91,19630,19632],{"className":19631},[98],[100,19633,19634],{"xmlns":102},[104,19635,19636,19640],{},[107,19637,19638],{},[113,19639,2249],{},[121,19641,2249],{"encoding":123},[91,19643,19645],{"className":19644,"ariaHidden":129},[128],[91,19646,19648,19651],{"className":19647},[133],[91,19649],{"className":19650,"style":261},[137],[91,19652,2249],{"className":19653},[142,212]," veces. Empecemos con los casos más sencillos:",[17,19656,19657],{},[33,19658,19659],{},"Potencia 2:",[17,19661,19663],{"className":19662},[645],[91,19664,19666,19716],{"className":19665},[94],[91,19667,19669],{"className":19668},[98],[100,19670,19671],{"xmlns":102},[104,19672,19673,19713],{},[107,19674,19675,19677,19679,19681,19683,19689,19691,19697,19699,19701,19703,19705,19707],{},[330,19676,1081],{"stretchy":9868},[113,19678,24],{},[330,19680,1368],{},[113,19682,19354],{},[1075,19684,19685,19687],{},[330,19686,1090],{"stretchy":9868},[117,19688,458],{},[330,19690,332],{},[1075,19692,19693,19695],{},[113,19694,24],{},[117,19696,458],{},[330,19698,1368],{},[117,19700,458],{},[113,19702,24],{},[113,19704,19354],{},[330,19706,1368],{},[1075,19708,19709,19711],{},[113,19710,19354],{},[117,19712,458],{},[121,19714,19715],{"encoding":123},"(a + b)^2 = a^2 + 2ab + b^2",[91,19717,19719,19740,19788,19832,19855],{"className":19718,"ariaHidden":129},[128],[91,19720,19722,19725,19728,19731,19734,19737],{"className":19721},[133],[91,19723],{"className":19724,"style":9889},[137],[91,19726,1081],{"className":19727},[146],[91,19729,24],{"className":19730},[142,212],[91,19732],{"className":19733,"style":2275},[420],[91,19735,1368],{"className":19736},[2279],[91,19738],{"className":19739,"style":2275},[420],[91,19741,19743,19747,19750,19779,19782,19785],{"className":19742},[133],[91,19744],{"className":19745,"style":19746},[137],"height:1.0641em;vertical-align:-0.25em;",[91,19748,19354],{"className":19749},[142,212],[91,19751,19753,19756],{"className":19752},[231],[91,19754,1090],{"className":19755},[231],[91,19757,19759],{"className":19758},[1209],[91,19760,19762],{"className":19761},[154],[91,19763,19765],{"className":19764},[159],[91,19766,19768],{"className":19767,"style":15860},[163],[91,19769,19770,19773],{"style":15606},[91,19771],{"className":19772,"style":1226},[171],[91,19774,19776],{"className":19775},[176,177,178,179],[91,19777,458],{"className":19778},[142,179],[91,19780],{"className":19781,"style":421},[420],[91,19783,332],{"className":19784},[425],[91,19786],{"className":19787,"style":421},[420],[91,19789,19791,19794,19823,19826,19829],{"className":19790},[133],[91,19792],{"className":19793,"style":19480},[137],[91,19795,19797,19800],{"className":19796},[142],[91,19798,24],{"className":19799},[142,212],[91,19801,19803],{"className":19802},[1209],[91,19804,19806],{"className":19805},[154],[91,19807,19809],{"className":19808},[159],[91,19810,19812],{"className":19811,"style":15860},[163],[91,19813,19814,19817],{"style":15606},[91,19815],{"className":19816,"style":1226},[171],[91,19818,19820],{"className":19819},[176,177,178,179],[91,19821,458],{"className":19822},[142,179],[91,19824],{"className":19825,"style":2275},[420],[91,19827,1368],{"className":19828},[2279],[91,19830],{"className":19831,"style":2275},[420],[91,19833,19835,19839,19842,19846,19849,19852],{"className":19834},[133],[91,19836],{"className":19837,"style":19838},[137],"height:0.7778em;vertical-align:-0.0833em;",[91,19840,458],{"className":19841},[142],[91,19843,19845],{"className":19844},[142,212],"ab",[91,19847],{"className":19848,"style":2275},[420],[91,19850,1368],{"className":19851},[2279],[91,19853],{"className":19854,"style":2275},[420],[91,19856,19858,19861],{"className":19857},[133],[91,19859],{"className":19860,"style":15860},[137],[91,19862,19864,19867],{"className":19863},[142],[91,19865,19354],{"className":19866},[142,212],[91,19868,19870],{"className":19869},[1209],[91,19871,19873],{"className":19872},[154],[91,19874,19876],{"className":19875},[159],[91,19877,19879],{"className":19878,"style":15860},[163],[91,19880,19881,19884],{"style":15606},[91,19882],{"className":19883,"style":1226},[171],[91,19885,19887],{"className":19886},[176,177,178,179],[91,19888,458],{"className":19889},[142,179],[17,19891,19892,19893,19948],{},"Los coeficientes son ",[91,19894,19896,19918],{"className":19895},[94],[91,19897,19899],{"className":19898},[98],[100,19900,19901],{"xmlns":102},[104,19902,19903,19915],{},[107,19904,19905,19907,19909,19911,19913],{},[117,19906,326],{},[330,19908,579],{"separator":129},[117,19910,458],{},[330,19912,579],{"separator":129},[117,19914,326],{},[121,19916,19917],{"encoding":123},"1, 2, 1",[91,19919,19921],{"className":19920,"ariaHidden":129},[128],[91,19922,19924,19927,19930,19933,19936,19939,19942,19945],{"className":19923},[133],[91,19925],{"className":19926,"style":567},[137],[91,19928,326],{"className":19929},[142],[91,19931,579],{"className":19932},[578],[91,19934],{"className":19935,"style":1504},[420],[91,19937,458],{"className":19938},[142],[91,19940,579],{"className":19941},[578],[91,19943],{"className":19944,"style":1504},[420],[91,19946,326],{"className":19947},[142]," — la fila 2 del triángulo.",[17,19950,19951],{},[33,19952,19953],{},"Potencia 3:",[17,19955,19957],{"className":19956},[645],[91,19958,19960,20026],{"className":19959},[94],[91,19961,19963],{"className":19962},[98],[100,19964,19965],{"xmlns":102},[104,19966,19967,20023],{},[107,19968,19969,19971,19973,19975,19977,19983,19985,19991,19993,19995,20001,20003,20005,20007,20009,20015,20017],{},[330,19970,1081],{"stretchy":9868},[113,19972,24],{},[330,19974,1368],{},[113,19976,19354],{},[1075,19978,19979,19981],{},[330,19980,1090],{"stretchy":9868},[117,19982,980],{},[330,19984,332],{},[1075,19986,19987,19989],{},[113,19988,24],{},[117,19990,980],{},[330,19992,1368],{},[117,19994,980],{},[1075,19996,19997,19999],{},[113,19998,24],{},[117,20000,458],{},[113,20002,19354],{},[330,20004,1368],{},[117,20006,980],{},[113,20008,24],{},[1075,20010,20011,20013],{},[113,20012,19354],{},[117,20014,458],{},[330,20016,1368],{},[1075,20018,20019,20021],{},[113,20020,19354],{},[117,20022,980],{},[121,20024,20025],{"encoding":123},"(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3",[91,20027,20029,20050,20097,20141,20191,20241],{"className":20028,"ariaHidden":129},[128],[91,20030,20032,20035,20038,20041,20044,20047],{"className":20031},[133],[91,20033],{"className":20034,"style":9889},[137],[91,20036,1081],{"className":20037},[146],[91,20039,24],{"className":20040},[142,212],[91,20042],{"className":20043,"style":2275},[420],[91,20045,1368],{"className":20046},[2279],[91,20048],{"className":20049,"style":2275},[420],[91,20051,20053,20056,20059,20088,20091,20094],{"className":20052},[133],[91,20054],{"className":20055,"style":19746},[137],[91,20057,19354],{"className":20058},[142,212],[91,20060,20062,20065],{"className":20061},[231],[91,20063,1090],{"className":20064},[231],[91,20066,20068],{"className":20067},[1209],[91,20069,20071],{"className":20070},[154],[91,20072,20074],{"className":20073},[159],[91,20075,20077],{"className":20076,"style":15860},[163],[91,20078,20079,20082],{"style":15606},[91,20080],{"className":20081,"style":1226},[171],[91,20083,20085],{"className":20084},[176,177,178,179],[91,20086,980],{"className":20087},[142,179],[91,20089],{"className":20090,"style":421},[420],[91,20092,332],{"className":20093},[425],[91,20095],{"className":20096,"style":421},[420],[91,20098,20100,20103,20132,20135,20138],{"className":20099},[133],[91,20101],{"className":20102,"style":19480},[137],[91,20104,20106,20109],{"className":20105},[142],[91,20107,24],{"className":20108},[142,212],[91,20110,20112],{"className":20111},[1209],[91,20113,20115],{"className":20114},[154],[91,20116,20118],{"className":20117},[159],[91,20119,20121],{"className":20120,"style":15860},[163],[91,20122,20123,20126],{"style":15606},[91,20124],{"className":20125,"style":1226},[171],[91,20127,20129],{"className":20128},[176,177,178,179],[91,20130,980],{"className":20131},[142,179],[91,20133],{"className":20134,"style":2275},[420],[91,20136,1368],{"className":20137},[2279],[91,20139],{"className":20140,"style":2275},[420],[91,20142,20144,20147,20150,20179,20182,20185,20188],{"className":20143},[133],[91,20145],{"className":20146,"style":19480},[137],[91,20148,980],{"className":20149},[142],[91,20151,20153,20156],{"className":20152},[142],[91,20154,24],{"className":20155},[142,212],[91,20157,20159],{"className":20158},[1209],[91,20160,20162],{"className":20161},[154],[91,20163,20165],{"className":20164},[159],[91,20166,20168],{"className":20167,"style":15860},[163],[91,20169,20170,20173],{"style":15606},[91,20171],{"className":20172,"style":1226},[171],[91,20174,20176],{"className":20175},[176,177,178,179],[91,20177,458],{"className":20178},[142,179],[91,20180,19354],{"className":20181},[142,212],[91,20183],{"className":20184,"style":2275},[420],[91,20186,1368],{"className":20187},[2279],[91,20189],{"className":20190,"style":2275},[420],[91,20192,20194,20197,20200,20203,20232,20235,20238],{"className":20193},[133],[91,20195],{"className":20196,"style":19480},[137],[91,20198,980],{"className":20199},[142],[91,20201,24],{"className":20202},[142,212],[91,20204,20206,20209],{"className":20205},[142],[91,20207,19354],{"className":20208},[142,212],[91,20210,20212],{"className":20211},[1209],[91,20213,20215],{"className":20214},[154],[91,20216,20218],{"className":20217},[159],[91,20219,20221],{"className":20220,"style":15860},[163],[91,20222,20223,20226],{"style":15606},[91,20224],{"className":20225,"style":1226},[171],[91,20227,20229],{"className":20228},[176,177,178,179],[91,20230,458],{"className":20231},[142,179],[91,20233],{"className":20234,"style":2275},[420],[91,20236,1368],{"className":20237},[2279],[91,20239],{"className":20240,"style":2275},[420],[91,20242,20244,20247],{"className":20243},[133],[91,20245],{"className":20246,"style":15860},[137],[91,20248,20250,20253],{"className":20249},[142],[91,20251,19354],{"className":20252},[142,212],[91,20254,20256],{"className":20255},[1209],[91,20257,20259],{"className":20258},[154],[91,20260,20262],{"className":20261},[159],[91,20263,20265],{"className":20264,"style":15860},[163],[91,20266,20267,20270],{"style":15606},[91,20268],{"className":20269,"style":1226},[171],[91,20271,20273],{"className":20272},[176,177,178,179],[91,20274,980],{"className":20275},[142,179],[17,20277,19892,20278,20345],{},[91,20279,20281,20306],{"className":20280},[94],[91,20282,20284],{"className":20283},[98],[100,20285,20286],{"xmlns":102},[104,20287,20288,20304],{},[107,20289,20290,20292,20294,20296,20298,20300,20302],{},[117,20291,326],{},[330,20293,579],{"separator":129},[117,20295,980],{},[330,20297,579],{"separator":129},[117,20299,980],{},[330,20301,579],{"separator":129},[117,20303,326],{},[121,20305,16006],{"encoding":123},[91,20307,20309],{"className":20308,"ariaHidden":129},[128],[91,20310,20312,20315,20318,20321,20324,20327,20330,20333,20336,20339,20342],{"className":20311},[133],[91,20313],{"className":20314,"style":567},[137],[91,20316,326],{"className":20317},[142],[91,20319,579],{"className":20320},[578],[91,20322],{"className":20323,"style":1504},[420],[91,20325,980],{"className":20326},[142],[91,20328,579],{"className":20329},[578],[91,20331],{"className":20332,"style":1504},[420],[91,20334,980],{"className":20335},[142],[91,20337,579],{"className":20338},[578],[91,20340],{"className":20341,"style":1504},[420],[91,20343,326],{"className":20344},[142]," — la fila 3.",[17,20347,20348],{},[33,20349,20350],{},"Potencia 5:",[17,20352,20354],{"className":20353},[645],[91,20355,20357,20455],{"className":20356},[94],[91,20358,20360],{"className":20359},[98],[100,20361,20362],{"xmlns":102},[104,20363,20364,20452],{},[107,20365,20366,20368,20370,20372,20374,20380,20382,20388,20390,20392,20398,20400,20402,20404,20410,20416,20418,20420,20426,20432,20434,20436,20438,20444,20446],{},[330,20367,1081],{"stretchy":9868},[113,20369,24],{},[330,20371,1368],{},[113,20373,19354],{},[1075,20375,20376,20378],{},[330,20377,1090],{"stretchy":9868},[117,20379,583],{},[330,20381,332],{},[1075,20383,20384,20386],{},[113,20385,24],{},[117,20387,583],{},[330,20389,1368],{},[117,20391,583],{},[1075,20393,20394,20396],{},[113,20395,24],{},[117,20397,119],{},[113,20399,19354],{},[330,20401,1368],{},[117,20403,15804],{},[1075,20405,20406,20408],{},[113,20407,24],{},[117,20409,980],{},[1075,20411,20412,20414],{},[113,20413,19354],{},[117,20415,458],{},[330,20417,1368],{},[117,20419,15804],{},[1075,20421,20422,20424],{},[113,20423,24],{},[117,20425,458],{},[1075,20427,20428,20430],{},[113,20429,19354],{},[117,20431,980],{},[330,20433,1368],{},[117,20435,583],{},[113,20437,24],{},[1075,20439,20440,20442],{},[113,20441,19354],{},[117,20443,119],{},[330,20445,1368],{},[1075,20447,20448,20450],{},[113,20449,19354],{},[117,20451,583],{},[121,20453,20454],{"encoding":123},"(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5",[91,20456,20458,20479,20526,20570,20620,20696,20772,20822],{"className":20457,"ariaHidden":129},[128],[91,20459,20461,20464,20467,20470,20473,20476],{"className":20460},[133],[91,20462],{"className":20463,"style":9889},[137],[91,20465,1081],{"className":20466},[146],[91,20468,24],{"className":20469},[142,212],[91,20471],{"className":20472,"style":2275},[420],[91,20474,1368],{"className":20475},[2279],[91,20477],{"className":20478,"style":2275},[420],[91,20480,20482,20485,20488,20517,20520,20523],{"className":20481},[133],[91,20483],{"className":20484,"style":19746},[137],[91,20486,19354],{"className":20487},[142,212],[91,20489,20491,20494],{"className":20490},[231],[91,20492,1090],{"className":20493},[231],[91,20495,20497],{"className":20496},[1209],[91,20498,20500],{"className":20499},[154],[91,20501,20503],{"className":20502},[159],[91,20504,20506],{"className":20505,"style":15860},[163],[91,20507,20508,20511],{"style":15606},[91,20509],{"className":20510,"style":1226},[171],[91,20512,20514],{"className":20513},[176,177,178,179],[91,20515,583],{"className":20516},[142,179],[91,20518],{"className":20519,"style":421},[420],[91,20521,332],{"className":20522},[425],[91,20524],{"className":20525,"style":421},[420],[91,20527,20529,20532,20561,20564,20567],{"className":20528},[133],[91,20530],{"className":20531,"style":19480},[137],[91,20533,20535,20538],{"className":20534},[142],[91,20536,24],{"className":20537},[142,212],[91,20539,20541],{"className":20540},[1209],[91,20542,20544],{"className":20543},[154],[91,20545,20547],{"className":20546},[159],[91,20548,20550],{"className":20549,"style":15860},[163],[91,20551,20552,20555],{"style":15606},[91,20553],{"className":20554,"style":1226},[171],[91,20556,20558],{"className":20557},[176,177,178,179],[91,20559,583],{"className":20560},[142,179],[91,20562],{"className":20563,"style":2275},[420],[91,20565,1368],{"className":20566},[2279],[91,20568],{"className":20569,"style":2275},[420],[91,20571,20573,20576,20579,20608,20611,20614,20617],{"className":20572},[133],[91,20574],{"className":20575,"style":19480},[137],[91,20577,583],{"className":20578},[142],[91,20580,20582,20585],{"className":20581},[142],[91,20583,24],{"className":20584},[142,212],[91,20586,20588],{"className":20587},[1209],[91,20589,20591],{"className":20590},[154],[91,20592,20594],{"className":20593},[159],[91,20595,20597],{"className":20596,"style":15860},[163],[91,20598,20599,20602],{"style":15606},[91,20600],{"className":20601,"style":1226},[171],[91,20603,20605],{"className":20604},[176,177,178,179],[91,20606,119],{"className":20607},[142,179],[91,20609,19354],{"className":20610},[142,212],[91,20612],{"className":20613,"style":2275},[420],[91,20615,1368],{"className":20616},[2279],[91,20618],{"className":20619,"style":2275},[420],[91,20621,20623,20626,20629,20658,20687,20690,20693],{"className":20622},[133],[91,20624],{"className":20625,"style":19480},[137],[91,20627,15804],{"className":20628},[142],[91,20630,20632,20635],{"className":20631},[142],[91,20633,24],{"className":20634},[142,212],[91,20636,20638],{"className":20637},[1209],[91,20639,20641],{"className":20640},[154],[91,20642,20644],{"className":20643},[159],[91,20645,20647],{"className":20646,"style":15860},[163],[91,20648,20649,20652],{"style":15606},[91,20650],{"className":20651,"style":1226},[171],[91,20653,20655],{"className":20654},[176,177,178,179],[91,20656,980],{"className":20657},[142,179],[91,20659,20661,20664],{"className":20660},[142],[91,20662,19354],{"className":20663},[142,212],[91,20665,20667],{"className":20666},[1209],[91,20668,20670],{"className":20669},[154],[91,20671,20673],{"className":20672},[159],[91,20674,20676],{"className":20675,"style":15860},[163],[91,20677,20678,20681],{"style":15606},[91,20679],{"className":20680,"style":1226},[171],[91,20682,20684],{"className":20683},[176,177,178,179],[91,20685,458],{"className":20686},[142,179],[91,20688],{"className":20689,"style":2275},[420],[91,20691,1368],{"className":20692},[2279],[91,20694],{"className":20695,"style":2275},[420],[91,20697,20699,20702,20705,20734,20763,20766,20769],{"className":20698},[133],[91,20700],{"className":20701,"style":19480},[137],[91,20703,15804],{"className":20704},[142],[91,20706,20708,20711],{"className":20707},[142],[91,20709,24],{"className":20710},[142,212],[91,20712,20714],{"className":20713},[1209],[91,20715,20717],{"className":20716},[154],[91,20718,20720],{"className":20719},[159],[91,20721,20723],{"className":20722,"style":15860},[163],[91,20724,20725,20728],{"style":15606},[91,20726],{"className":20727,"style":1226},[171],[91,20729,20731],{"className":20730},[176,177,178,179],[91,20732,458],{"className":20733},[142,179],[91,20735,20737,20740],{"className":20736},[142],[91,20738,19354],{"className":20739},[142,212],[91,20741,20743],{"className":20742},[1209],[91,20744,20746],{"className":20745},[154],[91,20747,20749],{"className":20748},[159],[91,20750,20752],{"className":20751,"style":15860},[163],[91,20753,20754,20757],{"style":15606},[91,20755],{"className":20756,"style":1226},[171],[91,20758,20760],{"className":20759},[176,177,178,179],[91,20761,980],{"className":20762},[142,179],[91,20764],{"className":20765,"style":2275},[420],[91,20767,1368],{"className":20768},[2279],[91,20770],{"className":20771,"style":2275},[420],[91,20773,20775,20778,20781,20784,20813,20816,20819],{"className":20774},[133],[91,20776],{"className":20777,"style":19480},[137],[91,20779,583],{"className":20780},[142],[91,20782,24],{"className":20783},[142,212],[91,20785,20787,20790],{"className":20786},[142],[91,20788,19354],{"className":20789},[142,212],[91,20791,20793],{"className":20792},[1209],[91,20794,20796],{"className":20795},[154],[91,20797,20799],{"className":20798},[159],[91,20800,20802],{"className":20801,"style":15860},[163],[91,20803,20804,20807],{"style":15606},[91,20805],{"className":20806,"style":1226},[171],[91,20808,20810],{"className":20809},[176,177,178,179],[91,20811,119],{"className":20812},[142,179],[91,20814],{"className":20815,"style":2275},[420],[91,20817,1368],{"className":20818},[2279],[91,20820],{"className":20821,"style":2275},[420],[91,20823,20825,20828],{"className":20824},[133],[91,20826],{"className":20827,"style":15860},[137],[91,20829,20831,20834],{"className":20830},[142],[91,20832,19354],{"className":20833},[142,212],[91,20835,20837],{"className":20836},[1209],[91,20838,20840],{"className":20839},[154],[91,20841,20843],{"className":20842},[159],[91,20844,20846],{"className":20845,"style":15860},[163],[91,20847,20848,20851],{"style":15606},[91,20849],{"className":20850,"style":1226},[171],[91,20852,20854],{"className":20853},[176,177,178,179],[91,20855,583],{"className":20856},[142,179],[17,20858,19892,20859,20953],{},[91,20860,20862,20896],{"className":20861},[94],[91,20863,20865],{"className":20864},[98],[100,20866,20867],{"xmlns":102},[104,20868,20869,20893],{},[107,20870,20871,20873,20875,20877,20879,20881,20883,20885,20887,20889,20891],{},[117,20872,326],{},[330,20874,579],{"separator":129},[117,20876,583],{},[330,20878,579],{"separator":129},[117,20880,15804],{},[330,20882,579],{"separator":129},[117,20884,15804],{},[330,20886,579],{"separator":129},[117,20888,583],{},[330,20890,579],{"separator":129},[117,20892,326],{},[121,20894,20895],{"encoding":123},"1, 5, 10, 10, 5, 1",[91,20897,20899],{"className":20898,"ariaHidden":129},[128],[91,20900,20902,20905,20908,20911,20914,20917,20920,20923,20926,20929,20932,20935,20938,20941,20944,20947,20950],{"className":20901},[133],[91,20903],{"className":20904,"style":567},[137],[91,20906,326],{"className":20907},[142],[91,20909,579],{"className":20910},[578],[91,20912],{"className":20913,"style":1504},[420],[91,20915,583],{"className":20916},[142],[91,20918,579],{"className":20919},[578],[91,20921],{"className":20922,"style":1504},[420],[91,20924,15804],{"className":20925},[142],[91,20927,579],{"className":20928},[578],[91,20930],{"className":20931,"style":1504},[420],[91,20933,15804],{"className":20934},[142],[91,20936,579],{"className":20937},[578],[91,20939],{"className":20940,"style":1504},[420],[91,20942,583],{"className":20943},[142],[91,20945,579],{"className":20946},[578],[91,20948],{"className":20949,"style":1504},[420],[91,20951,326],{"className":20952},[142]," — la fila 5.",[17,20955,20956,20957,20960,20961,11867],{},"¿Ves el patrón? ",[33,20958,20959],{},"Las filas del triángulo de Tartaglia son exactamente los coeficientes del desarrollo del binomio."," Esta relación se formaliza con el ",[33,20962,20963],{},"teorema del binomio",[17,20965,20967],{"className":20966},[645],[91,20968,20970,21047],{"className":20969},[94],[91,20971,20973],{"className":20972},[98],[100,20974,20975],{"xmlns":102},[104,20976,20977,21044],{},[107,20978,20979,20981,20983,20985,20987,20993,20995,21009,21021,21024,21036,21038],{},[330,20980,1081],{"stretchy":9868},[113,20982,24],{},[330,20984,1368],{},[113,20986,19354],{},[1075,20988,20989,20991],{},[330,20990,1090],{"stretchy":9868},[113,20992,2249],{},[330,20994,332],{},[7355,20996,20997,20999,21007],{},[330,20998,7359],{},[107,21000,21001,21003,21005],{},[113,21002,13539],{},[330,21004,332],{},[117,21006,571],{},[113,21008,2249],{},[107,21010,21011,21013,21019],{},[330,21012,1081],{"fence":129},[110,21014,21015,21017],{"linethickness":4373},[113,21016,2249],{},[113,21018,13539],{},[330,21020,1090],{"fence":129},[659,21022,21023],{}," ",[1075,21025,21026,21028],{},[113,21027,24],{},[107,21029,21030,21032,21034],{},[113,21031,2249],{},[330,21033,1405],{},[113,21035,13539],{},[659,21037,21023],{},[1075,21039,21040,21042],{},[113,21041,19354],{},[113,21043,13539],{},[121,21045,21046],{"encoding":123},"(a+b)^n = \\sum_{k=0}^{n} \\binom{n}{k} \\, a^{n-k} \\, b^k",[91,21048,21050,21071,21118],{"className":21049,"ariaHidden":129},[128],[91,21051,21053,21056,21059,21062,21065,21068],{"className":21052},[133],[91,21054],{"className":21055,"style":9889},[137],[91,21057,1081],{"className":21058},[146],[91,21060,24],{"className":21061},[142,212],[91,21063],{"className":21064,"style":2275},[420],[91,21066,1368],{"className":21067},[2279],[91,21069],{"className":21070,"style":2275},[420],[91,21072,21074,21077,21080,21109,21112,21115],{"className":21073},[133],[91,21075],{"className":21076,"style":9889},[137],[91,21078,19354],{"className":21079},[142,212],[91,21081,21083,21086],{"className":21082},[231],[91,21084,1090],{"className":21085},[231],[91,21087,21089],{"className":21088},[1209],[91,21090,21092],{"className":21091},[154],[91,21093,21095],{"className":21094},[159],[91,21096,21098],{"className":21097,"style":15585},[163],[91,21099,21100,21103],{"style":15606},[91,21101],{"className":21102,"style":1226},[171],[91,21104,21106],{"className":21105},[176,177,178,179],[91,21107,2249],{"className":21108},[142,212,179],[91,21110],{"className":21111,"style":421},[420],[91,21113,332],{"className":21114},[425],[91,21116],{"className":21117,"style":421},[420],[91,21119,21121,21124,21187,21190,21256,21259,21298,21301],{"className":21120},[133],[91,21122],{"className":21123,"style":13554},[137],[91,21125,21127,21130],{"className":21126},[7454],[91,21128,7359],{"className":21129,"style":7460},[7454,7458,7459],[91,21131,21133],{"className":21132},[1209],[91,21134,21136,21179],{"className":21135},[154,155],[91,21137,21139,21176],{"className":21138},[159],[91,21140,21142,21162],{"className":21141,"style":7473},[163],[91,21143,21144,21147],{"style":7476},[91,21145],{"className":21146,"style":1226},[171],[91,21148,21150],{"className":21149},[176,177,178,179],[91,21151,21153,21156,21159],{"className":21152},[142,179],[91,21154,13539],{"className":21155,"style":13591},[142,212,179],[91,21157,332],{"className":21158},[425,179],[91,21160,571],{"className":21161},[142,179],[91,21163,21164,21167],{"style":7497},[91,21165],{"className":21166,"style":1226},[171],[91,21168,21170],{"className":21169},[176,177,178,179],[91,21171,21173],{"className":21172},[142,179],[91,21174,2249],{"className":21175},[142,212,179],[91,21177,218],{"className":21178},[217],[91,21180,21182],{"className":21181},[159],[91,21183,21185],{"className":21184,"style":7519},[163],[91,21186],{},[91,21188],{"className":21189,"style":1504},[420],[91,21191,21193,21199,21250],{"className":21192},[142],[91,21194,21196],{"className":21195,"style":1126},[146,1125],[91,21197,1081],{"className":21198},[1130,1131],[91,21200,21202],{"className":21201},[110],[91,21203,21205,21242],{"className":21204},[154,155],[91,21206,21208,21239],{"className":21207},[159],[91,21209,21211,21225],{"className":21210,"style":13576},[163],[91,21212,21213,21216],{"style":4555},[91,21214],{"className":21215,"style":1226},[171],[91,21217,21219],{"className":21218},[176,177,178,179],[91,21220,21222],{"className":21221},[142,179],[91,21223,13539],{"className":21224,"style":13591},[142,212,179],[91,21226,21227,21230],{"style":4570},[91,21228],{"className":21229,"style":1226},[171],[91,21231,21233],{"className":21232},[176,177,178,179],[91,21234,21236],{"className":21235},[142,179],[91,21237,2249],{"className":21238},[142,212,179],[91,21240,218],{"className":21241},[217],[91,21243,21245],{"className":21244},[159],[91,21246,21248],{"className":21247,"style":225},[163],[91,21249],{},[91,21251,21253],{"className":21252,"style":1126},[231,1125],[91,21254,1090],{"className":21255},[1130,1131],[91,21257],{"className":21258,"style":1504},[420],[91,21260,21262,21265],{"className":21261},[142],[91,21263,24],{"className":21264},[142,212],[91,21266,21268],{"className":21267},[1209],[91,21269,21271],{"className":21270},[154],[91,21272,21274],{"className":21273},[159],[91,21275,21278],{"className":21276,"style":21277},[163],"height:0.8491em;",[91,21279,21280,21283],{"style":15606},[91,21281],{"className":21282,"style":1226},[171],[91,21284,21286],{"className":21285},[176,177,178,179],[91,21287,21289,21292,21295],{"className":21288},[142,179],[91,21290,2249],{"className":21291},[142,212,179],[91,21293,1405],{"className":21294},[2279,179],[91,21296,13539],{"className":21297,"style":13591},[142,212,179],[91,21299],{"className":21300,"style":1504},[420],[91,21302,21304,21307],{"className":21303},[142],[91,21305,19354],{"className":21306},[142,212],[91,21308,21310],{"className":21309},[1209],[91,21311,21313],{"className":21312},[154],[91,21314,21316],{"className":21315},[159],[91,21317,21319],{"className":21318,"style":21277},[163],[91,21320,21321,21324],{"style":15606},[91,21322],{"className":21323,"style":1226},[171],[91,21325,21327],{"className":21326},[176,177,178,179],[91,21328,13539],{"className":21329,"style":13591},[142,212,179],[17,21331,21332,21333,21362,21363,14217,21414,21465,21466,21565],{},"El símbolo ",[91,21334,21336,21350],{"className":21335},[94],[91,21337,21339],{"className":21338},[98],[100,21340,21341],{"xmlns":102},[104,21342,21343,21347],{},[107,21344,21345],{},[330,21346,7359],{},[121,21348,21349],{"encoding":123},"\\sum",[91,21351,21353],{"className":21352,"ariaHidden":129},[128],[91,21354,21356,21359],{"className":21355},[133],[91,21357],{"className":21358,"style":9889},[137],[91,21360,7359],{"className":21361,"style":7460},[7454,7458,7459]," (sigma) simplemente significa «suma todos los términos desde ",[91,21364,21366,21384],{"className":21365},[94],[91,21367,21369],{"className":21368},[98],[100,21370,21371],{"xmlns":102},[104,21372,21373,21381],{},[107,21374,21375,21377,21379],{},[113,21376,13539],{},[330,21378,332],{},[117,21380,571],{},[121,21382,21383],{"encoding":123},"k = 0",[91,21385,21387,21405],{"className":21386,"ariaHidden":129},[128],[91,21388,21390,21393,21396,21399,21402],{"className":21389},[133],[91,21391],{"className":21392,"style":7078},[137],[91,21394,13539],{"className":21395,"style":13591},[142,212],[91,21397],{"className":21398,"style":421},[420],[91,21400,332],{"className":21401},[425],[91,21403],{"className":21404,"style":421},[420],[91,21406,21408,21411],{"className":21407},[133],[91,21409],{"className":21410,"style":435},[137],[91,21412,571],{"className":21413},[142],[91,21415,21417,21435],{"className":21416},[94],[91,21418,21420],{"className":21419},[98],[100,21421,21422],{"xmlns":102},[104,21423,21424,21432],{},[107,21425,21426,21428,21430],{},[113,21427,13539],{},[330,21429,332],{},[113,21431,2249],{},[121,21433,21434],{"encoding":123},"k = n",[91,21436,21438,21456],{"className":21437,"ariaHidden":129},[128],[91,21439,21441,21444,21447,21450,21453],{"className":21440},[133],[91,21442],{"className":21443,"style":7078},[137],[91,21445,13539],{"className":21446,"style":13591},[142,212],[91,21448],{"className":21449,"style":421},[420],[91,21451,332],{"className":21452},[425],[91,21454],{"className":21455,"style":421},[420],[91,21457,21459,21462],{"className":21458},[133],[91,21460],{"className":21461,"style":261},[137],[91,21463,2249],{"className":21464},[142,212],"». Cada término tiene un coeficiente binomial ",[91,21467,21469,21490],{"className":21468},[94],[91,21470,21472],{"className":21471},[98],[100,21473,21474],{"xmlns":102},[104,21475,21476,21488],{},[107,21477,21478,21480,21486],{},[330,21479,1081],{"fence":129},[110,21481,21482,21484],{"linethickness":4373},[113,21483,2249],{},[113,21485,13539],{},[330,21487,1090],{"fence":129},[121,21489,13544],{"encoding":123},[91,21491,21493],{"className":21492,"ariaHidden":129},[128],[91,21494,21496,21499],{"className":21495},[133],[91,21497],{"className":21498,"style":13554},[137],[91,21500,21502,21508,21559],{"className":21501},[142],[91,21503,21505],{"className":21504,"style":1126},[146,1125],[91,21506,1081],{"className":21507},[1130,1131],[91,21509,21511],{"className":21510},[110],[91,21512,21514,21551],{"className":21513},[154,155],[91,21515,21517,21548],{"className":21516},[159],[91,21518,21520,21534],{"className":21519,"style":13576},[163],[91,21521,21522,21525],{"style":4555},[91,21523],{"className":21524,"style":1226},[171],[91,21526,21528],{"className":21527},[176,177,178,179],[91,21529,21531],{"className":21530},[142,179],[91,21532,13539],{"className":21533,"style":13591},[142,212,179],[91,21535,21536,21539],{"style":4570},[91,21537],{"className":21538,"style":1226},[171],[91,21540,21542],{"className":21541},[176,177,178,179],[91,21543,21545],{"className":21544},[142,179],[91,21546,2249],{"className":21547},[142,212,179],[91,21549,218],{"className":21550},[217],[91,21552,21554],{"className":21553},[159],[91,21555,21557],{"className":21556,"style":225},[163],[91,21558],{},[91,21560,21562],{"className":21561,"style":1126},[231,1125],[91,21563,1090],{"className":21564},[1130,1131]," que viene directamente del triángulo.",[635,21567,21569],{"id":21568},"el-puente-entre-álgebra-y-probabilidad","El puente entre álgebra y probabilidad",[17,21571,21572,21573,21601,21602,584,21604,21601,21632,11867],{},"Ahora viene lo fascinante. Sustituye ",[91,21574,21576,21589],{"className":21575},[94],[91,21577,21579],{"className":21578},[98],[100,21580,21581],{"xmlns":102},[104,21582,21583,21587],{},[107,21584,21585],{},[113,21586,24],{},[121,21588,24],{"encoding":123},[91,21590,21592],{"className":21591,"ariaHidden":129},[128],[91,21593,21595,21598],{"className":21594},[133],[91,21596],{"className":21597,"style":261},[137],[91,21599,24],{"className":21600},[142,212]," por la probabilidad de obtener ",[33,21603,15342],{},[91,21605,21607,21620],{"className":21606},[94],[91,21608,21610],{"className":21609},[98],[100,21611,21612],{"xmlns":102},[104,21613,21614,21618],{},[107,21615,21616],{},[113,21617,19354],{},[121,21619,19354],{"encoding":123},[91,21621,21623],{"className":21622,"ariaHidden":129},[128],[91,21624,21626,21629],{"className":21625},[133],[91,21627],{"className":21628,"style":7078},[137],[91,21630,19354],{"className":21631},[142,212],[33,21633,15346],{},[17,21635,21637],{"className":21636},[645],[91,21638,21640,21696],{"className":21639},[94],[91,21641,21643],{"className":21642},[98],[100,21644,21645],{"xmlns":102},[104,21646,21647,21693],{},[107,21648,21649,21651,21653,21655,21657,21659,21661,21663,21669,21671,21673,21675,21677,21679,21681,21683,21685,21687],{},[113,21650,24],{},[330,21652,332],{},[113,21654,16295],{},[330,21656,1081],{"stretchy":9868},[659,21658,15342],{},[330,21660,1090],{"stretchy":9868},[330,21662,332],{},[110,21664,21665,21667],{},[117,21666,326],{},[117,21668,458],{},[330,21670,579],{"separator":129},[420,21672],{"width":18067},[113,21674,19354],{},[330,21676,332],{},[113,21678,16295],{},[330,21680,1081],{"stretchy":9868},[659,21682,15346],{},[330,21684,1090],{"stretchy":9868},[330,21686,332],{},[110,21688,21689,21691],{},[117,21690,326],{},[117,21692,458],{},[121,21694,21695],{"encoding":123},"a = P(\\text{cara}) = \\frac{1}{2}, \\quad b = P(\\text{cruz}) = \\frac{1}{2}",[91,21697,21699,21717,21747,21842,21872],{"className":21698,"ariaHidden":129},[128],[91,21700,21702,21705,21708,21711,21714],{"className":21701},[133],[91,21703],{"className":21704,"style":261},[137],[91,21706,24],{"className":21707},[142,212],[91,21709],{"className":21710,"style":421},[420],[91,21712,332],{"className":21713},[425],[91,21715],{"className":21716,"style":421},[420],[91,21718,21720,21723,21726,21729,21735,21738,21741,21744],{"className":21719},[133],[91,21721],{"className":21722,"style":9889},[137],[91,21724,16295],{"className":21725,"style":16321},[142,212],[91,21727,1081],{"className":21728},[146],[91,21730,21732],{"className":21731},[142,678],[91,21733,15342],{"className":21734},[142],[91,21736,1090],{"className":21737},[231],[91,21739],{"className":21740,"style":421},[420],[91,21742,332],{"className":21743},[425],[91,21745],{"className":21746,"style":421},[420],[91,21748,21750,21753,21821,21824,21827,21830,21833,21836,21839],{"className":21749},[133],[91,21751],{"className":21752,"style":347},[137],[91,21754,21756,21759,21818],{"className":21755},[142],[91,21757],{"className":21758},[146,147],[91,21760,21762],{"className":21761},[110],[91,21763,21765,21810],{"className":21764},[154,155],[91,21766,21768,21807],{"className":21767},[159],[91,21769,21771,21785,21793],{"className":21770,"style":366},[163],[91,21772,21773,21776],{"style":167},[91,21774],{"className":21775,"style":172},[171],[91,21777,21779],{"className":21778},[176,177,178,179],[91,21780,21782],{"className":21781},[142,179],[91,21783,458],{"className":21784},[142,179],[91,21786,21787,21790],{"style":188},[91,21788],{"className":21789,"style":172},[171],[91,21791],{"className":21792,"style":196},[195],[91,21794,21795,21798],{"style":199},[91,21796],{"className":21797,"style":172},[171],[91,21799,21801],{"className":21800},[176,177,178,179],[91,21802,21804],{"className":21803},[142,179],[91,21805,326],{"className":21806},[142,179],[91,21808,218],{"className":21809},[217],[91,21811,21813],{"className":21812},[159],[91,21814,21816],{"className":21815,"style":225},[163],[91,21817],{},[91,21819],{"className":21820},[231,147],[91,21822,579],{"className":21823},[578],[91,21825],{"className":21826,"style":18104},[420],[91,21828],{"className":21829,"style":1504},[420],[91,21831,19354],{"className":21832},[142,212],[91,21834],{"className":21835,"style":421},[420],[91,21837,332],{"className":21838},[425],[91,21840],{"className":21841,"style":421},[420],[91,21843,21845,21848,21851,21854,21860,21863,21866,21869],{"className":21844},[133],[91,21846],{"className":21847,"style":9889},[137],[91,21849,16295],{"className":21850,"style":16321},[142,212],[91,21852,1081],{"className":21853},[146],[91,21855,21857],{"className":21856},[142,678],[91,21858,15346],{"className":21859},[142],[91,21861,1090],{"className":21862},[231],[91,21864],{"className":21865,"style":421},[420],[91,21867,332],{"className":21868},[425],[91,21870],{"className":21871,"style":421},[420],[91,21873,21875,21878],{"className":21874},[133],[91,21876],{"className":21877,"style":347},[137],[91,21879,21881,21884,21943],{"className":21880},[142],[91,21882],{"className":21883},[146,147],[91,21885,21887],{"className":21886},[110],[91,21888,21890,21935],{"className":21889},[154,155],[91,21891,21893,21932],{"className":21892},[159],[91,21894,21896,21910,21918],{"className":21895,"style":366},[163],[91,21897,21898,21901],{"style":167},[91,21899],{"className":21900,"style":172},[171],[91,21902,21904],{"className":21903},[176,177,178,179],[91,21905,21907],{"className":21906},[142,179],[91,21908,458],{"className":21909},[142,179],[91,21911,21912,21915],{"style":188},[91,21913],{"className":21914,"style":172},[171],[91,21916],{"className":21917,"style":196},[195],[91,21919,21920,21923],{"style":199},[91,21921],{"className":21922,"style":172},[171],[91,21924,21926],{"className":21925},[176,177,178,179],[91,21927,21929],{"className":21928},[142,179],[91,21930,326],{"className":21931},[142,179],[91,21933,218],{"className":21934},[217],[91,21936,21938],{"className":21937},[159],[91,21939,21941],{"className":21940,"style":225},[163],[91,21942],{},[91,21944],{"className":21945},[231,147],[17,21947,21948],{},"El binomio se convierte en:",[17,21950,21952],{"className":21951},[645],[91,21953,21955,22093],{"className":21954},[94],[91,21956,21958],{"className":21957},[98],[100,21959,21960],{"xmlns":102},[104,21961,21962,22090],{},[107,21963,21964,21988,21990,22004,22016,22038,22054,22056,22070],{},[1075,21965,21966,21986],{},[107,21967,21968,21970,21976,21978,21984],{},[330,21969,1081],{"fence":129},[110,21971,21972,21974],{},[117,21973,326],{},[117,21975,458],{},[330,21977,1368],{},[110,21979,21980,21982],{},[117,21981,326],{},[117,21983,458],{},[330,21985,1090],{"fence":129},[113,21987,2249],{},[330,21989,332],{},[7355,21991,21992,21994,22002],{},[330,21993,7359],{},[107,21995,21996,21998,22000],{},[113,21997,13539],{},[330,21999,332],{},[117,22001,571],{},[113,22003,2249],{},[107,22005,22006,22008,22014],{},[330,22007,1081],{"fence":129},[110,22009,22010,22012],{"linethickness":4373},[113,22011,2249],{},[113,22013,13539],{},[330,22015,1090],{"fence":129},[1075,22017,22018,22030],{},[107,22019,22020,22022,22028],{},[330,22021,1081],{"fence":129},[110,22023,22024,22026],{},[117,22025,326],{},[117,22027,458],{},[330,22029,1090],{"fence":129},[107,22031,22032,22034,22036],{},[113,22033,2249],{},[330,22035,1405],{},[113,22037,13539],{},[1075,22039,22040,22052],{},[107,22041,22042,22044,22050],{},[330,22043,1081],{"fence":129},[110,22045,22046,22048],{},[117,22047,326],{},[117,22049,458],{},[330,22051,1090],{"fence":129},[113,22053,13539],{},[330,22055,332],{},[7355,22057,22058,22060,22068],{},[330,22059,7359],{},[107,22061,22062,22064,22066],{},[113,22063,13539],{},[330,22065,332],{},[117,22067,571],{},[113,22069,2249],{},[110,22071,22072,22084],{},[107,22073,22074,22076,22082],{},[330,22075,1081],{"fence":129},[110,22077,22078,22080],{"linethickness":4373},[113,22079,2249],{},[113,22081,13539],{},[330,22083,1090],{"fence":129},[1075,22085,22086,22088],{},[117,22087,458],{},[113,22089,2249],{},[121,22091,22092],{"encoding":123},"\\left(\\frac{1}{2} + \\frac{1}{2}\\right)^n = \\sum_{k=0}^{n} \\binom{n}{k} \\left(\\frac{1}{2}\\right)^{n-k} \\left(\\frac{1}{2}\\right)^k = \\sum_{k=0}^{n} \\frac{\\binom{n}{k}}{2^n}",[91,22094,22096,22299,22681],{"className":22095,"ariaHidden":129},[128],[91,22097,22099,22103,22290,22293,22296],{"className":22098},[133],[91,22100],{"className":22101,"style":22102},[137],"height:1.2543em;vertical-align:-0.35em;",[91,22104,22106,22266],{"className":22105},[1118],[91,22107,22109,22115,22183,22186,22189,22192,22260],{"className":22108},[1118],[91,22110,22112],{"className":22111,"style":1126},[146,1125],[91,22113,1081],{"className":22114},[1130,1131],[91,22116,22118,22121,22180],{"className":22117},[142],[91,22119],{"className":22120},[146,147],[91,22122,22124],{"className":22123},[110],[91,22125,22127,22172],{"className":22126},[154,155],[91,22128,22130,22169],{"className":22129},[159],[91,22131,22133,22147,22155],{"className":22132,"style":366},[163],[91,22134,22135,22138],{"style":167},[91,22136],{"className":22137,"style":172},[171],[91,22139,22141],{"className":22140},[176,177,178,179],[91,22142,22144],{"className":22143},[142,179],[91,22145,458],{"className":22146},[142,179],[91,22148,22149,22152],{"style":188},[91,22150],{"className":22151,"style":172},[171],[91,22153],{"className":22154,"style":196},[195],[91,22156,22157,22160],{"style":199},[91,22158],{"className":22159,"style":172},[171],[91,22161,22163],{"className":22162},[176,177,178,179],[91,22164,22166],{"className":22165},[142,179],[91,22167,326],{"className":22168},[142,179],[91,22170,218],{"className":22171},[217],[91,22173,22175],{"className":22174},[159],[91,22176,22178],{"className":22177,"style":225},[163],[91,22179],{},[91,22181],{"className":22182},[231,147],[91,22184],{"className":22185,"style":2275},[420],[91,22187,1368],{"className":22188},[2279],[91,22190],{"className":22191,"style":2275},[420],[91,22193,22195,22198,22257],{"className":22194},[142],[91,22196],{"className":22197},[146,147],[91,22199,22201],{"className":22200},[110],[91,22202,22204,22249],{"className":22203},[154,155],[91,22205,22207,22246],{"className":22206},[159],[91,22208,22210,22224,22232],{"className":22209,"style":366},[163],[91,22211,22212,22215],{"style":167},[91,22213],{"className":22214,"style":172},[171],[91,22216,22218],{"className":22217},[176,177,178,179],[91,22219,22221],{"className":22220},[142,179],[91,22222,458],{"className":22223},[142,179],[91,22225,22226,22229],{"style":188},[91,22227],{"className":22228,"style":172},[171],[91,22230],{"className":22231,"style":196},[195],[91,22233,22234,22237],{"style":199},[91,22235],{"className":22236,"style":172},[171],[91,22238,22240],{"className":22239},[176,177,178,179],[91,22241,22243],{"className":22242},[142,179],[91,22244,326],{"className":22245},[142,179],[91,22247,218],{"className":22248},[217],[91,22250,22252],{"className":22251},[159],[91,22253,22255],{"className":22254,"style":225},[163],[91,22256],{},[91,22258],{"className":22259},[231,147],[91,22261,22263],{"className":22262,"style":1126},[231,1125],[91,22264,1090],{"className":22265},[1130,1131],[91,22267,22269],{"className":22268},[1209],[91,22270,22272],{"className":22271},[154],[91,22273,22275],{"className":22274},[159],[91,22276,22279],{"className":22277,"style":22278},[163],"height:0.9043em;",[91,22280,22281,22284],{"style":1222},[91,22282],{"className":22283,"style":1226},[171],[91,22285,22287],{"className":22286},[176,177,178,179],[91,22288,2249],{"className":22289},[142,212,179],[91,22291],{"className":22292,"style":421},[420],[91,22294,332],{"className":22295},[425],[91,22297],{"className":22298,"style":421},[420],[91,22300,22302,22306,22369,22372,22438,22441,22560,22563,22672,22675,22678],{"className":22301},[133],[91,22303],{"className":22304,"style":22305},[137],"height:1.439em;vertical-align:-0.35em;",[91,22307,22309,22312],{"className":22308},[7454],[91,22310,7359],{"className":22311,"style":7460},[7454,7458,7459],[91,22313,22315],{"className":22314},[1209],[91,22316,22318,22361],{"className":22317},[154,155],[91,22319,22321,22358],{"className":22320},[159],[91,22322,22324,22344],{"className":22323,"style":7473},[163],[91,22325,22326,22329],{"style":7476},[91,22327],{"className":22328,"style":1226},[171],[91,22330,22332],{"className":22331},[176,177,178,179],[91,22333,22335,22338,22341],{"className":22334},[142,179],[91,22336,13539],{"className":22337,"style":13591},[142,212,179],[91,22339,332],{"className":22340},[425,179],[91,22342,571],{"className":22343},[142,179],[91,22345,22346,22349],{"style":7497},[91,22347],{"className":22348,"style":1226},[171],[91,22350,22352],{"className":22351},[176,177,178,179],[91,22353,22355],{"className":22354},[142,179],[91,22356,2249],{"className":22357},[142,212,179],[91,22359,218],{"className":22360},[217],[91,22362,22364],{"className":22363},[159],[91,22365,22367],{"className":22366,"style":7519},[163],[91,22368],{},[91,22370],{"className":22371,"style":1504},[420],[91,22373,22375,22381,22432],{"className":22374},[142],[91,22376,22378],{"className":22377,"style":1126},[146,1125],[91,22379,1081],{"className":22380},[1130,1131],[91,22382,22384],{"className":22383},[110],[91,22385,22387,22424],{"className":22386},[154,155],[91,22388,22390,22421],{"className":22389},[159],[91,22391,22393,22407],{"className":22392,"style":13576},[163],[91,22394,22395,22398],{"style":4555},[91,22396],{"className":22397,"style":1226},[171],[91,22399,22401],{"className":22400},[176,177,178,179],[91,22402,22404],{"className":22403},[142,179],[91,22405,13539],{"className":22406,"style":13591},[142,212,179],[91,22408,22409,22412],{"style":4570},[91,22410],{"className":22411,"style":1226},[171],[91,22413,22415],{"className":22414},[176,177,178,179],[91,22416,22418],{"className":22417},[142,179],[91,22419,2249],{"className":22420},[142,212,179],[91,22422,218],{"className":22423},[217],[91,22425,22427],{"className":22426},[159],[91,22428,22430],{"className":22429,"style":225},[163],[91,22431],{},[91,22433,22435],{"className":22434,"style":1126},[231,1125],[91,22436,1090],{"className":22437},[1130,1131],[91,22439],{"className":22440,"style":1504},[420],[91,22442,22444,22527],{"className":22443},[1118],[91,22445,22447,22453,22521],{"className":22446},[1118],[91,22448,22450],{"className":22449,"style":1126},[146,1125],[91,22451,1081],{"className":22452},[1130,1131],[91,22454,22456,22459,22518],{"className":22455},[142],[91,22457],{"className":22458},[146,147],[91,22460,22462],{"className":22461},[110],[91,22463,22465,22510],{"className":22464},[154,155],[91,22466,22468,22507],{"className":22467},[159],[91,22469,22471,22485,22493],{"className":22470,"style":366},[163],[91,22472,22473,22476],{"style":167},[91,22474],{"className":22475,"style":172},[171],[91,22477,22479],{"className":22478},[176,177,178,179],[91,22480,22482],{"className":22481},[142,179],[91,22483,458],{"className":22484},[142,179],[91,22486,22487,22490],{"style":188},[91,22488],{"className":22489,"style":172},[171],[91,22491],{"className":22492,"style":196},[195],[91,22494,22495,22498],{"style":199},[91,22496],{"className":22497,"style":172},[171],[91,22499,22501],{"className":22500},[176,177,178,179],[91,22502,22504],{"className":22503},[142,179],[91,22505,326],{"className":22506},[142,179],[91,22508,218],{"className":22509},[217],[91,22511,22513],{"className":22512},[159],[91,22514,22516],{"className":22515,"style":225},[163],[91,22517],{},[91,22519],{"className":22520},[231,147],[91,22522,22524],{"className":22523,"style":1126},[231,1125],[91,22525,1090],{"className":22526},[1130,1131],[91,22528,22530],{"className":22529},[1209],[91,22531,22533],{"className":22532},[154],[91,22534,22536],{"className":22535},[159],[91,22537,22540],{"className":22538,"style":22539},[163],"height:1.089em;",[91,22541,22542,22545],{"style":1222},[91,22543],{"className":22544,"style":1226},[171],[91,22546,22548],{"className":22547},[176,177,178,179],[91,22549,22551,22554,22557],{"className":22550},[142,179],[91,22552,2249],{"className":22553},[142,212,179],[91,22555,1405],{"className":22556},[2279,179],[91,22558,13539],{"className":22559,"style":13591},[142,212,179],[91,22561],{"className":22562,"style":1504},[420],[91,22564,22566,22649],{"className":22565},[1118],[91,22567,22569,22575,22643],{"className":22568},[1118],[91,22570,22572],{"className":22571,"style":1126},[146,1125],[91,22573,1081],{"className":22574},[1130,1131],[91,22576,22578,22581,22640],{"className":22577},[142],[91,22579],{"className":22580},[146,147],[91,22582,22584],{"className":22583},[110],[91,22585,22587,22632],{"className":22586},[154,155],[91,22588,22590,22629],{"className":22589},[159],[91,22591,22593,22607,22615],{"className":22592,"style":366},[163],[91,22594,22595,22598],{"style":167},[91,22596],{"className":22597,"style":172},[171],[91,22599,22601],{"className":22600},[176,177,178,179],[91,22602,22604],{"className":22603},[142,179],[91,22605,458],{"className":22606},[142,179],[91,22608,22609,22612],{"style":188},[91,22610],{"className":22611,"style":172},[171],[91,22613],{"className":22614,"style":196},[195],[91,22616,22617,22620],{"style":199},[91,22618],{"className":22619,"style":172},[171],[91,22621,22623],{"className":22622},[176,177,178,179],[91,22624,22626],{"className":22625},[142,179],[91,22627,326],{"className":22628},[142,179],[91,22630,218],{"className":22631},[217],[91,22633,22635],{"className":22634},[159],[91,22636,22638],{"className":22637,"style":225},[163],[91,22639],{},[91,22641],{"className":22642},[231,147],[91,22644,22646],{"className":22645,"style":1126},[231,1125],[91,22647,1090],{"className":22648},[1130,1131],[91,22650,22652],{"className":22651},[1209],[91,22653,22655],{"className":22654},[154],[91,22656,22658],{"className":22657},[159],[91,22659,22661],{"className":22660,"style":22539},[163],[91,22662,22663,22666],{"style":1222},[91,22664],{"className":22665,"style":1226},[171],[91,22667,22669],{"className":22668},[176,177,178,179],[91,22670,13539],{"className":22671,"style":13591},[142,212,179],[91,22673],{"className":22674,"style":421},[420],[91,22676,332],{"className":22677},[425],[91,22679],{"className":22680,"style":421},[420],[91,22682,22684,22687,22750,22753],{"className":22683},[133],[91,22685],{"className":22686,"style":11357},[137],[91,22688,22690,22693],{"className":22689},[7454],[91,22691,7359],{"className":22692,"style":7460},[7454,7458,7459],[91,22694,22696],{"className":22695},[1209],[91,22697,22699,22742],{"className":22698},[154,155],[91,22700,22702,22739],{"className":22701},[159],[91,22703,22705,22725],{"className":22704,"style":7473},[163],[91,22706,22707,22710],{"style":7476},[91,22708],{"className":22709,"style":1226},[171],[91,22711,22713],{"className":22712},[176,177,178,179],[91,22714,22716,22719,22722],{"className":22715},[142,179],[91,22717,13539],{"className":22718,"style":13591},[142,212,179],[91,22720,332],{"className":22721},[425,179],[91,22723,571],{"className":22724},[142,179],[91,22726,22727,22730],{"style":7497},[91,22728],{"className":22729,"style":1226},[171],[91,22731,22733],{"className":22732},[176,177,178,179],[91,22734,22736],{"className":22735},[142,179],[91,22737,2249],{"className":22738},[142,212,179],[91,22740,218],{"className":22741},[217],[91,22743,22745],{"className":22744},[159],[91,22746,22748],{"className":22747,"style":7519},[163],[91,22749],{},[91,22751],{"className":22752,"style":1504},[420],[91,22754,22756,22759,22913],{"className":22755},[142],[91,22757],{"className":22758},[146,147],[91,22760,22762],{"className":22761},[110],[91,22763,22765,22905],{"className":22764},[154,155],[91,22766,22768,22902],{"className":22767},[159],[91,22769,22771,22811,22819],{"className":22770,"style":10300},[163],[91,22772,22773,22776],{"style":167},[91,22774],{"className":22775,"style":172},[171],[91,22777,22779],{"className":22778},[176,177,178,179],[91,22780,22782],{"className":22781},[142,179],[91,22783,22785,22788],{"className":22784},[142,179],[91,22786,458],{"className":22787},[142,179],[91,22789,22791],{"className":22790},[1209],[91,22792,22794],{"className":22793},[154],[91,22795,22797],{"className":22796},[159],[91,22798,22800],{"className":22799,"style":11077},[163],[91,22801,22802,22805],{"style":7576},[91,22803],{"className":22804,"style":7580},[171],[91,22806,22808],{"className":22807},[176,7584,1131,179],[91,22809,2249],{"className":22810},[142,212,179],[91,22812,22813,22816],{"style":188},[91,22814],{"className":22815,"style":172},[171],[91,22817],{"className":22818,"style":196},[195],[91,22820,22821,22824],{"style":10334},[91,22822],{"className":22823,"style":172},[171],[91,22825,22827],{"className":22826},[176,177,178,179],[91,22828,22830],{"className":22829},[142,179],[91,22831,22833,22842,22893],{"className":22832},[142,179],[91,22834,22836],{"className":22835,"style":10351},[146,176,7584,10350,179,1125],[91,22837,22839],{"className":22838},[1130,1131,179],[91,22840,1081],{"className":22841},[179],[91,22843,22845],{"className":22844},[110],[91,22846,22848,22885],{"className":22847},[154,155],[91,22849,22851,22882],{"className":22850},[159],[91,22852,22854,22868],{"className":22853,"style":17978},[163],[91,22855,22856,22859],{"style":10373},[91,22857],{"className":22858,"style":7580},[171],[91,22860,22862],{"className":22861},[176,7584,1131,179],[91,22863,22865],{"className":22864},[142,179],[91,22866,13539],{"className":22867,"style":13591},[142,212,179],[91,22869,22870,22873],{"style":10388},[91,22871],{"className":22872,"style":7580},[171],[91,22874,22876],{"className":22875},[176,7584,1131,179],[91,22877,22879],{"className":22878},[142,179],[91,22880,2249],{"className":22881},[142,212,179],[91,22883,218],{"className":22884},[217],[91,22886,22888],{"className":22887},[159],[91,22889,22891],{"className":22890,"style":10413},[163],[91,22892],{},[91,22894,22896],{"className":22895,"style":10351},[231,176,7584,10350,179,1125],[91,22897,22899],{"className":22898},[1130,1131,179],[91,22900,1090],{"className":22901},[179],[91,22903,218],{"className":22904},[217],[91,22906,22908],{"className":22907},[159],[91,22909,22911],{"className":22910,"style":225},[163],[91,22912],{},[91,22914],{"className":22915},[231,147],[17,22917,22918,22919,23000,23001,23208,23209,23237,23238,23290],{},"El lado izquierdo es simplemente ",[91,22920,22922,22944],{"className":22921},[94],[91,22923,22925],{"className":22924},[98],[100,22926,22927],{"xmlns":102},[104,22928,22929,22941],{},[107,22930,22931,22937,22939],{},[1075,22932,22933,22935],{},[117,22934,326],{},[113,22936,2249],{},[330,22938,332],{},[117,22940,326],{},[121,22942,22943],{"encoding":123},"1^n = 1",[91,22945,22947,22991],{"className":22946,"ariaHidden":129},[128],[91,22948,22950,22953,22982,22985,22988],{"className":22949},[133],[91,22951],{"className":22952,"style":15585},[137],[91,22954,22956,22959],{"className":22955},[142],[91,22957,326],{"className":22958},[142],[91,22960,22962],{"className":22961},[1209],[91,22963,22965],{"className":22964},[154],[91,22966,22968],{"className":22967},[159],[91,22969,22971],{"className":22970,"style":15585},[163],[91,22972,22973,22976],{"style":15606},[91,22974],{"className":22975,"style":1226},[171],[91,22977,22979],{"className":22978},[176,177,178,179],[91,22980,2249],{"className":22981},[142,212,179],[91,22983],{"className":22984,"style":421},[420],[91,22986,332],{"className":22987},[425],[91,22989],{"className":22990,"style":421},[420],[91,22992,22994,22997],{"className":22993},[133],[91,22995],{"className":22996,"style":435},[137],[91,22998,326],{"className":22999},[142],": la certeza total. El lado derecho es la suma de todas las probabilidades individuales. Y cada término ",[91,23002,23004,23036],{"className":23003},[94],[91,23005,23007],{"className":23006},[98],[100,23008,23009],{"xmlns":102},[104,23010,23011,23033],{},[107,23012,23013],{},[110,23014,23015,23027],{},[107,23016,23017,23019,23025],{},[330,23018,1081],{"fence":129},[110,23020,23021,23023],{"linethickness":4373},[113,23022,2249],{},[113,23024,13539],{},[330,23026,1090],{"fence":129},[1075,23028,23029,23031],{},[117,23030,458],{},[113,23032,2249],{},[121,23034,23035],{"encoding":123},"\\frac{\\binom{n}{k}}{2^n}",[91,23037,23039],{"className":23038,"ariaHidden":129},[128],[91,23040,23042,23045],{"className":23041},[133],[91,23043],{"className":23044,"style":11357},[137],[91,23046,23048,23051,23205],{"className":23047},[142],[91,23049],{"className":23050},[146,147],[91,23052,23054],{"className":23053},[110],[91,23055,23057,23197],{"className":23056},[154,155],[91,23058,23060,23194],{"className":23059},[159],[91,23061,23063,23103,23111],{"className":23062,"style":10300},[163],[91,23064,23065,23068],{"style":167},[91,23066],{"className":23067,"style":172},[171],[91,23069,23071],{"className":23070},[176,177,178,179],[91,23072,23074],{"className":23073},[142,179],[91,23075,23077,23080],{"className":23076},[142,179],[91,23078,458],{"className":23079},[142,179],[91,23081,23083],{"className":23082},[1209],[91,23084,23086],{"className":23085},[154],[91,23087,23089],{"className":23088},[159],[91,23090,23092],{"className":23091,"style":11077},[163],[91,23093,23094,23097],{"style":7576},[91,23095],{"className":23096,"style":7580},[171],[91,23098,23100],{"className":23099},[176,7584,1131,179],[91,23101,2249],{"className":23102},[142,212,179],[91,23104,23105,23108],{"style":188},[91,23106],{"className":23107,"style":172},[171],[91,23109],{"className":23110,"style":196},[195],[91,23112,23113,23116],{"style":10334},[91,23114],{"className":23115,"style":172},[171],[91,23117,23119],{"className":23118},[176,177,178,179],[91,23120,23122],{"className":23121},[142,179],[91,23123,23125,23134,23185],{"className":23124},[142,179],[91,23126,23128],{"className":23127,"style":10351},[146,176,7584,10350,179,1125],[91,23129,23131],{"className":23130},[1130,1131,179],[91,23132,1081],{"className":23133},[179],[91,23135,23137],{"className":23136},[110],[91,23138,23140,23177],{"className":23139},[154,155],[91,23141,23143,23174],{"className":23142},[159],[91,23144,23146,23160],{"className":23145,"style":17978},[163],[91,23147,23148,23151],{"style":10373},[91,23149],{"className":23150,"style":7580},[171],[91,23152,23154],{"className":23153},[176,7584,1131,179],[91,23155,23157],{"className":23156},[142,179],[91,23158,13539],{"className":23159,"style":13591},[142,212,179],[91,23161,23162,23165],{"style":10388},[91,23163],{"className":23164,"style":7580},[171],[91,23166,23168],{"className":23167},[176,7584,1131,179],[91,23169,23171],{"className":23170},[142,179],[91,23172,2249],{"className":23173},[142,212,179],[91,23175,218],{"className":23176},[217],[91,23178,23180],{"className":23179},[159],[91,23181,23183],{"className":23182,"style":10413},[163],[91,23184],{},[91,23186,23188],{"className":23187,"style":10351},[231,176,7584,10350,179,1125],[91,23189,23191],{"className":23190},[1130,1131,179],[91,23192,1090],{"className":23193},[179],[91,23195,218],{"className":23196},[217],[91,23198,23200],{"className":23199},[159],[91,23201,23203],{"className":23202,"style":225},[163],[91,23204],{},[91,23206],{"className":23207},[231,147]," es exactamente la probabilidad de obtener ",[91,23210,23212,23225],{"className":23211},[94],[91,23213,23215],{"className":23214},[98],[100,23216,23217],{"xmlns":102},[104,23218,23219,23223],{},[107,23220,23221],{},[113,23222,13539],{},[121,23224,13539],{"encoding":123},[91,23226,23228],{"className":23227,"ariaHidden":129},[128],[91,23229,23231,23234],{"className":23230},[133],[91,23232],{"className":23233,"style":7078},[137],[91,23235,13539],{"className":23236,"style":13591},[142,212]," cruces (o equivalentemente, ",[91,23239,23241,23259],{"className":23240},[94],[91,23242,23244],{"className":23243},[98],[100,23245,23246],{"xmlns":102},[104,23247,23248,23256],{},[107,23249,23250,23252,23254],{},[113,23251,2249],{},[330,23253,1405],{},[113,23255,13539],{},[121,23257,23258],{"encoding":123},"n - k",[91,23260,23262,23281],{"className":23261,"ariaHidden":129},[128],[91,23263,23265,23269,23272,23275,23278],{"className":23264},[133],[91,23266],{"className":23267,"style":23268},[137],"height:0.6667em;vertical-align:-0.0833em;",[91,23270,2249],{"className":23271},[142,212],[91,23273],{"className":23274,"style":2275},[420],[91,23276,1405],{"className":23277},[2279],[91,23279],{"className":23280,"style":2275},[420],[91,23282,23284,23287],{"className":23283},[133],[91,23285],{"className":23286,"style":7078},[137],[91,23288,13539],{"className":23289,"style":13591},[142,212]," caras).",[17,23292,23293],{},"Comprobémoslo con 5 lanzamientos:",[17,23295,23297],{"className":23296},[645],[91,23298,23300,23352],{"className":23299},[94],[91,23301,23303],{"className":23302},[98],[100,23304,23305],{"xmlns":102},[104,23306,23307,23349],{},[107,23308,23309,23337,23339,23345,23347],{},[110,23310,23311,23335],{},[107,23312,23313,23315,23317,23319,23321,23323,23325,23327,23329,23331,23333],{},[117,23314,326],{},[330,23316,1368],{},[117,23318,583],{},[330,23320,1368],{},[117,23322,15804],{},[330,23324,1368],{},[117,23326,15804],{},[330,23328,1368],{},[117,23330,583],{},[330,23332,1368],{},[117,23334,326],{},[117,23336,1899],{},[330,23338,332],{},[110,23340,23341,23343],{},[117,23342,1899],{},[117,23344,1899],{},[330,23346,332],{},[117,23348,326],{},[121,23350,23351],{"encoding":123},"\\frac{1 + 5 + 10 + 10 + 5 + 1}{32} = \\frac{32}{32} = 1",[91,23353,23355,23468,23551],{"className":23354,"ariaHidden":129},[128],[91,23356,23358,23361,23459,23462,23465],{"className":23357},[133],[91,23359],{"className":23360,"style":347},[137],[91,23362,23364,23367,23456],{"className":23363},[142],[91,23365],{"className":23366},[146,147],[91,23368,23370],{"className":23369},[110],[91,23371,23373,23448],{"className":23372},[154,155],[91,23374,23376,23445],{"className":23375},[159],[91,23377,23379,23393,23401],{"className":23378,"style":366},[163],[91,23380,23381,23384],{"style":167},[91,23382],{"className":23383,"style":172},[171],[91,23385,23387],{"className":23386},[176,177,178,179],[91,23388,23390],{"className":23389},[142,179],[91,23391,1899],{"className":23392},[142,179],[91,23394,23395,23398],{"style":188},[91,23396],{"className":23397,"style":172},[171],[91,23399],{"className":23400,"style":196},[195],[91,23402,23403,23406],{"style":199},[91,23404],{"className":23405,"style":172},[171],[91,23407,23409],{"className":23408},[176,177,178,179],[91,23410,23412,23415,23418,23421,23424,23427,23430,23433,23436,23439,23442],{"className":23411},[142,179],[91,23413,326],{"className":23414},[142,179],[91,23416,1368],{"className":23417},[2279,179],[91,23419,583],{"className":23420},[142,179],[91,23422,1368],{"className":23423},[2279,179],[91,23425,15804],{"className":23426},[142,179],[91,23428,1368],{"className":23429},[2279,179],[91,23431,15804],{"className":23432},[142,179],[91,23434,1368],{"className":23435},[2279,179],[91,23437,583],{"className":23438},[142,179],[91,23440,1368],{"className":23441},[2279,179],[91,23443,326],{"className":23444},[142,179],[91,23446,218],{"className":23447},[217],[91,23449,23451],{"className":23450},[159],[91,23452,23454],{"className":23453,"style":225},[163],[91,23455],{},[91,23457],{"className":23458},[231,147],[91,23460],{"className":23461,"style":421},[420],[91,23463,332],{"className":23464},[425],[91,23466],{"className":23467,"style":421},[420],[91,23469,23471,23474,23542,23545,23548],{"className":23470},[133],[91,23472],{"className":23473,"style":347},[137],[91,23475,23477,23480,23539],{"className":23476},[142],[91,23478],{"className":23479},[146,147],[91,23481,23483],{"className":23482},[110],[91,23484,23486,23531],{"className":23485},[154,155],[91,23487,23489,23528],{"className":23488},[159],[91,23490,23492,23506,23514],{"className":23491,"style":366},[163],[91,23493,23494,23497],{"style":167},[91,23495],{"className":23496,"style":172},[171],[91,23498,23500],{"className":23499},[176,177,178,179],[91,23501,23503],{"className":23502},[142,179],[91,23504,1899],{"className":23505},[142,179],[91,23507,23508,23511],{"style":188},[91,23509],{"className":23510,"style":172},[171],[91,23512],{"className":23513,"style":196},[195],[91,23515,23516,23519],{"style":199},[91,23517],{"className":23518,"style":172},[171],[91,23520,23522],{"className":23521},[176,177,178,179],[91,23523,23525],{"className":23524},[142,179],[91,23526,1899],{"className":23527},[142,179],[91,23529,218],{"className":23530},[217],[91,23532,23534],{"className":23533},[159],[91,23535,23537],{"className":23536,"style":225},[163],[91,23538],{},[91,23540],{"className":23541},[231,147],[91,23543],{"className":23544,"style":421},[420],[91,23546,332],{"className":23547},[425],[91,23549],{"className":23550,"style":421},[420],[91,23552,23554,23557],{"className":23553},[133],[91,23555],{"className":23556,"style":435},[137],[91,23558,326],{"className":23559},[142],[17,23561,23562,23563,23622,23623,23651,23652,40],{},"Cada número de la fila 5 dividido entre ",[91,23564,23566,23584],{"className":23565},[94],[91,23567,23569],{"className":23568},[98],[100,23570,23571],{"xmlns":102},[104,23572,23573,23581],{},[107,23574,23575],{},[1075,23576,23577,23579],{},[117,23578,458],{},[117,23580,583],{},[121,23582,23583],{"encoding":123},"2^5",[91,23585,23587],{"className":23586,"ariaHidden":129},[128],[91,23588,23590,23593],{"className":23589},[133],[91,23591],{"className":23592,"style":15860},[137],[91,23594,23596,23599],{"className":23595},[142],[91,23597,458],{"className":23598},[142],[91,23600,23602],{"className":23601},[1209],[91,23603,23605],{"className":23604},[154],[91,23606,23608],{"className":23607},[159],[91,23609,23611],{"className":23610,"style":15860},[163],[91,23612,23613,23616],{"style":15606},[91,23614],{"className":23615,"style":1226},[171],[91,23617,23619],{"className":23618},[176,177,178,179],[91,23620,583],{"className":23621},[142,179]," nos da una probabilidad, y juntas suman ",[91,23624,23626,23639],{"className":23625},[94],[91,23627,23629],{"className":23628},[98],[100,23630,23631],{"xmlns":102},[104,23632,23633,23637],{},[107,23634,23635],{},[117,23636,326],{},[121,23638,326],{"encoding":123},[91,23640,23642],{"className":23641,"ariaHidden":129},[128],[91,23643,23645,23648],{"className":23644},[133],[91,23646],{"className":23647,"style":435},[137],[91,23649,326],{"className":23650},[142],". El triángulo, la combinatoria y el álgebra no son tres historias diferentes: ",[33,23653,23654],{},"son la misma historia contada en tres idiomas distintos",[12,23656,12892],{"id":12891},[17,23658,23659],{},"A lo largo de este artículo hemos seguido un hilo que conecta tres ramas de las matemáticas a través de un mismo objeto:",[3601,23661,23662,23667,23672],{},[287,23663,15329,23664,23666],{},[33,23665,15332],{}," nos dice de cuántas maneras podemos organizar un conjunto de resultados.",[287,23668,4300,23669,23671],{},[33,23670,19329],{}," del binomio genera esas mismas cantidades como coeficientes de una fórmula.",[287,23673,15329,23674,23676],{},[33,23675,6521],{}," transforma esos números en predicciones sobre el mundo real.",[17,23678,23679,23680,23682],{},"Y el ",[33,23681,13194],{}," es el objeto que las une: una tabla visual donde las tres disciplinas convergen.",[17,23684,23685],{},"Lo que empezó como un triángulo para contar combinaciones — dibujado en la China del siglo XI — termina describiendo con precisión qué ocurre cuando lanzas una moneda cinco, diez o cien veces. Una estructura abstracta creada para contar objetos resulta describir fenómenos aleatorios.",[17,23687,23688],{},"Y hay un detalle más que merece una mención. A medida que el número de lanzamientos crece, las filas del triángulo se parecen cada vez más a la curva de campana de Gauss: esa curva ubicua que aparece en la estadística, la física y la biología. La razón profunda de esa convergencia es otra historia, pero el hecho de que un triángulo de sumas tan simples contenga la semilla de una de las distribuciones más importantes de la ciencia es, como mínimo, hermoso.",[17,23690,23691],{},"Las matemáticas tienen esa cualidad única: revelan patrones ocultos detrás de lo que parece puro azar. Un lanzamiento de moneda es impredecible. Diez lanzamientos ya insinúan un orden. Cien lanzamientos dibujan una curva casi perfecta. Y todo estaba escrito, desde el principio, en un triángulo.",[17,23693,12999,23694,23697],{},[24,23695,13003],{"href":631,"rel":23696},[81]," y observa la combinatoria en acción.",{"title":51,"searchDepth":52,"depth":52,"links":23699},[23700,23701,23704,23705,23708,23709,23712],{"id":13187,"depth":52,"text":13188},{"id":13215,"depth":52,"text":13216,"children":23702},[23703],{"id":13508,"depth":13010,"text":13509},{"id":15325,"depth":52,"text":15326},{"id":15821,"depth":52,"text":15822,"children":23706},[23707],{"id":16271,"depth":13010,"text":16272},{"id":18043,"depth":52,"text":18044},{"id":19322,"depth":52,"text":19323,"children":23710},[23711],{"id":21568,"depth":13010,"text":21569},{"id":12891,"depth":52,"text":12892},"2026-05-24T10:00:00Z","Descubre cómo el triángulo de Tartaglia conecta la combinatoria, el álgebra y la probabilidad — explicado paso a paso con lanzamientos de moneda.",{"excerpt":23716},{"type":9,"value":23717},[23718,23720],[71,23719,13176],{"id":13181},[17,23721,13184],{},"\u002Fblog\u002Fpascals-triangle-and-probability.es",{"title":13176,"description":23714},"triangulo-de-tartaglia-combinatoria-y-probabilidad","blog\u002Fpascals-triangle-and-probability.es","qZQyHpms2FZH6Ff6w11iThobO-PzvEaP5Ib4ymGhG00",{"id":23728,"title":23729,"_locale":7,"body":23730,"date":24150,"description":24151,"extension":56,"meta":24152,"navigation":58,"path":24159,"published":58,"seo":24160,"slug":24161,"stem":24162,"__hash__":24163},"blog\u002Fblog\u002Fheads-or-tails.es.md","Cara o Cruz — La Guía Completa sobre el Lanzamiento de Moneda",{"type":9,"value":23731,"toc":24142},[23732,23735,23738,23742,23748,23762,23769,23773,23780,23865,23950,24023,24035,24053,24057,24064,24075,24078,24082,24088,24094,24100,24106,24110,24117,24120,24123,24127,24133],[71,23733,23729],{"id":23734},"cara-o-cruz-la-guía-completa-sobre-el-lanzamiento-de-moneda",[17,23736,23737],{},"\"¿Cara o cruz?\" Tres palabras que han resuelto disputas, iniciado juegos y tomado decisiones durante miles de años. Pero ¿de dónde viene esa frase? ¿Y es la moneda realmente justa?",[12,23739,23741],{"id":23740},"de-dónde-vienen-cara-y-cruz","¿De dónde vienen \"cara\" y \"cruz\"?",[17,23743,23744,23747],{},[33,23745,23746],{},"Cara"," es sencillo: las monedas han tenido retratos desde la antigüedad. Las monedas griegas y romanas mostraban el perfil de un gobernante, deidad o emperador — una cara literal.",[17,23749,23750,23753,23754,23757,23758,23761],{},[33,23751,23752],{},"Cruz"," tiene una explicación más local. En la tradición hispánica, el reverso de muchas monedas mostraba una cruz — el símbolo religioso que marcaba el \"otro lado\" de la moneda. Así quedó establecida la dicotomía cara\u002Fcruz que usamos hoy. En latín, el juego se llamaba ",[267,23755,23756],{},"caput aut navis"," (\"cabeza o barco\"), porque el reverso mostraba la proa de una nave. En inglés moderno se dice ",[267,23759,23760],{},"heads or tails"," (\"cabeza o cola\"), refiriéndose a los animales representados en el reverso de las monedas británicas.",[17,23763,23764,23765,23768],{},"La versión romana evolucionó hasta el juego que Julio César hizo famoso: cuando las monedas empezaron a llevar su retrato, el juego pasó a llamarse ",[267,23766,23767],{},"capita aut navia"," (\"cabeza o barco\") en su honor.",[12,23770,23772],{"id":23771},"las-matemáticas-del-lanzamiento-de-moneda","Las matemáticas del lanzamiento de moneda",[17,23774,23775,23776,23779],{},"En teoría de la probabilidad, el lanzamiento de una moneda justa es el ejemplo por excelencia de un ",[33,23777,23778],{},"ensayo de Bernoulli",": un experimento aleatorio con exactamente dos resultados igualmente probables (1 ó 0). La probabilidad de cara es 0,5, la probabilidad de cruz es 0,5, y cada lanzamiento es independiente de todos los demás. En notación matemática, se representa como:",[17,23781,23783],{"className":23782},[645],[91,23784,23786,23814],{"className":23785},[94],[91,23787,23789],{"className":23788},[98],[100,23790,23791],{"xmlns":102},[104,23792,23793,23811],{},[107,23794,23795,23797,23799,23801,23803,23805,23807,23809],{},[113,23796,16295],{},[330,23798,1081],{"stretchy":9868},[659,23800,15342],{},[330,23802,1090],{"stretchy":9868},[330,23804,332],{},[117,23806,571],{},[330,23808,579],{"separator":129},[117,23810,583],{},[121,23812,23813],{"encoding":123},"P(\\text{cara}) = 0,5",[91,23815,23817,23847],{"className":23816,"ariaHidden":129},[128],[91,23818,23820,23823,23826,23829,23835,23838,23841,23844],{"className":23819},[133],[91,23821],{"className":23822,"style":9889},[137],[91,23824,16295],{"className":23825,"style":16321},[142,212],[91,23827,1081],{"className":23828},[146],[91,23830,23832],{"className":23831},[142,678],[91,23833,15342],{"className":23834},[142],[91,23836,1090],{"className":23837},[231],[91,23839],{"className":23840,"style":421},[420],[91,23842,332],{"className":23843},[425],[91,23845],{"className":23846,"style":421},[420],[91,23848,23850,23853,23856,23859,23862],{"className":23849},[133],[91,23851],{"className":23852,"style":567},[137],[91,23854,571],{"className":23855},[142],[91,23857,579],{"className":23858},[578],[91,23860],{"className":23861,"style":1504},[420],[91,23863,583],{"className":23864},[142],[17,23866,23868],{"className":23867},[645],[91,23869,23871,23899],{"className":23870},[94],[91,23872,23874],{"className":23873},[98],[100,23875,23876],{"xmlns":102},[104,23877,23878,23896],{},[107,23879,23880,23882,23884,23886,23888,23890,23892,23894],{},[113,23881,16295],{},[330,23883,1081],{"stretchy":9868},[659,23885,15346],{},[330,23887,1090],{"stretchy":9868},[330,23889,332],{},[117,23891,571],{},[330,23893,579],{"separator":129},[117,23895,583],{},[121,23897,23898],{"encoding":123},"P(\\text{cruz}) = 0,5",[91,23900,23902,23932],{"className":23901,"ariaHidden":129},[128],[91,23903,23905,23908,23911,23914,23920,23923,23926,23929],{"className":23904},[133],[91,23906],{"className":23907,"style":9889},[137],[91,23909,16295],{"className":23910,"style":16321},[142,212],[91,23912,1081],{"className":23913},[146],[91,23915,23917],{"className":23916},[142,678],[91,23918,15346],{"className":23919},[142],[91,23921,1090],{"className":23922},[231],[91,23924],{"className":23925,"style":421},[420],[91,23927,332],{"className":23928},[425],[91,23930],{"className":23931,"style":421},[420],[91,23933,23935,23938,23941,23944,23947],{"className":23934},[133],[91,23936],{"className":23937,"style":567},[137],[91,23939,571],{"className":23940},[142],[91,23942,579],{"className":23943},[578],[91,23945],{"className":23946,"style":1504},[420],[91,23948,583],{"className":23949},[142],[17,23951,23953],{"className":23952},[645],[91,23954,23956,23981],{"className":23955},[94],[91,23957,23959],{"className":23958},[98],[100,23960,23961],{"xmlns":102},[104,23962,23963,23978],{},[107,23964,23965,23967,23969,23972,23974,23976],{},[113,23966,16295],{},[330,23968,1081],{"stretchy":9868},[659,23970,23971],{},"cara o cruz",[330,23973,1090],{"stretchy":9868},[330,23975,332],{},[117,23977,326],{},[121,23979,23980],{"encoding":123},"P(\\text{cara o cruz}) = 1",[91,23982,23984,24014],{"className":23983,"ariaHidden":129},[128],[91,23985,23987,23990,23993,23996,24002,24005,24008,24011],{"className":23986},[133],[91,23988],{"className":23989,"style":9889},[137],[91,23991,16295],{"className":23992,"style":16321},[142,212],[91,23994,1081],{"className":23995},[146],[91,23997,23999],{"className":23998},[142,678],[91,24000,23971],{"className":24001},[142],[91,24003,1090],{"className":24004},[231],[91,24006],{"className":24007,"style":421},[420],[91,24009,332],{"className":24010},[425],[91,24012],{"className":24013,"style":421},[420],[91,24015,24017,24020],{"className":24016},[133],[91,24018],{"className":24019,"style":435},[137],[91,24021,326],{"className":24022},[142],[17,24024,24025,24026,24029,24030,40],{},"Esa independencia tiene una implicación famosa: si acabas de sacar diez caras seguidas, el undécimo lanzamiento sigue siendo 50\u002F50. La moneda no tiene memoria. Creer lo contrario es la ",[33,24027,24028],{},"Falacia del Jugador"," — ",[24,24031,24034],{"href":24032,"rel":24033},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fla-estadistica-inesperada-de-los-lanzamientos-de-moneda",[81],"explorada en profundidad aquí",[17,24036,24037,24038,24040,24041,24044,24045,584,24047,24049,24050,24052],{},"El número de caras esperado en ",[267,24039,2249],{}," lanzamientos sigue una ",[33,24042,24043],{},"distribución binomial"," con parámetros ",[267,24046,2249],{},[267,24048,17],{}," = 0,5. Para ",[267,24051,2249],{}," grande, esto se aproxima a una distribución normal — razón por la que los lanzamientos de moneda se usan para enseñar el Teorema Central del Límite, el cual explica que la suma de muchas variables aleatorias independientes sigue una distribución normal (forma de campana).",[12,24054,24056],{"id":24055},"es-realmente-justo-un-lanzamiento-de-moneda","¿Es realmente justo un lanzamiento de moneda?",[17,24058,24059,24060,24063],{},"No del todo. El estudio del matemático de Stanford Persi Diaconis mencionado en este blog demostró que una moneda lanzada de forma natural cae sobre la misma cara con la que empezó aproximadamente el ",[33,24061,24062],{},"51% de las veces",". El sesgo viene de:",[3601,24065,24066,24069,24072],{},[287,24067,24068],{},"La moneda pasa ligeramente más tiempo rotando en su orientación inicial antes de que la gravedad tome el control",[287,24070,24071],{},"La mecánica del pulgar que consistentemente favorece una dirección",[287,24073,24074],{},"La resistencia del aire que afecta la cara que empieza hacia arriba",[17,24076,24077],{},"Un sesgo del 1% suena pequeño. En un solo lanzamiento, es irrelevante. En miles de lanzamientos en un torneo o estudio estadístico, importa. Aunque parezca que los simuladores online basados en física no heredan la mecánica humana de lanzamiento, arrastran parte de ese sesgo y no pueden garantizar un resultado 50\u002F50, aunque se repita el lanzamiento muchas veces.",[12,24079,24081],{"id":24080},"lanzamientos-de-moneda-famosos-en-la-historia","Lanzamientos de moneda famosos en la historia",[17,24083,24084,24087],{},[33,24085,24086],{},"La Copa Davis",": El protocolo oficial de la ITF exige un lanzamiento de moneda antes de cada eliminatoria para decidir quién elige pista o servicio. Millones de partidos han comenzado así.",[17,24089,24090,24093],{},[33,24091,24092],{},"Los Hermanos Wright (1903)",": Orville y Wilbur Wright supuestamente lanzaron una moneda para decidir quién intentaría el primer vuelo motorizado en Kitty Hawk. Wilbur ganó el sorteo pero su primer intento falló. El segundo intento de Orville, tres días después, tuvo éxito.",[17,24095,24096,24099],{},[33,24097,24098],{},"El Caucus Demócrata de Iowa (2015)",": Varios distritos empataron entre Hillary Clinton y Bernie Sanders. ¿El desempate? Un lanzamiento de moneda. Clinton supuestamente ganó los seis lanzamientos — una secuencia con una probabilidad de aproximadamente el 1,6%, improbable pero completamente posible.",[17,24101,24102,24105],{},[33,24103,24104],{},"El fútbol",": La FIFA exige lanzamientos de moneda antes de los partidos internacionales. El equipo perdedor elige el extremo que defenderá; el capitán ganador decide si hace el saque inicial o lo cede al rival.",[12,24107,24109],{"id":24108},"por-qué-confiamos-en-los-lanzamientos-de-moneda","Por qué confiamos en los lanzamientos de moneda",[17,24111,24112,24113,24116],{},"Hay algo filosóficamente interesante en el lanzamiento de moneda: se acepta como justo precisamente porque ",[267,24114,24115],{},"nadie lo controla",". La moneda es un árbitro que no tiene interés en el resultado.",[17,24118,24119],{},"Por eso los lanzamientos de moneda se usan para resolver disputas en lugar de, por ejemplo, piedra-papel-tijera (que puede verse influenciado por la psicología y las señales). El elemento aleatorio elimina la agencia humana — y con ella, la posibilidad de queja.",[17,24121,24122],{},"Un lanzamiento de moneda online cumple la misma función. Cuando ambas partes pueden ver el lanzamiento en tiempo real, el resultado tiene el mismo peso social que un lanzamiento físico.",[12,24124,24126],{"id":24125},"conclusión","Conclusión",[17,24128,24129,24132],{},[33,24130,24131],{},"Cara o cruz"," es una de las herramientas de aleatorización más antiguas de la historia humana — simple, fiable y, como resulta, no perfectamente justa en el mundo real. Un simulador de física online te acerca más al ideal teórico de lo que podría hacerlo un lanzamiento humano.",[17,24134,24135,24136,24141],{},"¿Listo para lanzar? ",[24,24137,24140],{"href":24138,"rel":24139},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fplay",[81],"Prueba cara o cruz online"," — física real, resultado instantáneo.",{"title":51,"searchDepth":52,"depth":52,"links":24143},[24144,24145,24146,24147,24148,24149],{"id":23740,"depth":52,"text":23741},{"id":23771,"depth":52,"text":23772},{"id":24055,"depth":52,"text":24056},{"id":24080,"depth":52,"text":24081},{"id":24108,"depth":52,"text":24109},{"id":24125,"depth":52,"text":24126},"2026-05-22T12:00:00Z","¿Por qué se llama cara o cruz? Historia, matemáticas y la sorprendente ciencia del lanzamiento de moneda — de Julio César a la física moderna.",{"excerpt":24153},{"type":9,"value":24154},[24155,24157],[71,24156,23729],{"id":23734},[17,24158,23737],{},"\u002Fblog\u002Fheads-or-tails.es",{"title":23729,"description":24151},"cara-o-cruz","blog\u002Fheads-or-tails.es","d01guGbrghT6q7EPXImYAgI3Mnc7bMm9ywHZ4KkDC2U",1779926066352]