[{"data":1,"prerenderedAt":10726},["ShallowReactive",2],{"blog-post-es-triangulo-de-tartaglia-combinatoria-y-probabilidad":3,"locale-switcher-blog-es-triangulo-de-tartaglia-combinatoria-y-probabilidad":10723,"blog-siblings-es-triangulo-de-tartaglia-combinatoria-y-probabilidad":10725},{"id":4,"title":5,"_locale":6,"body":7,"date":10707,"description":10708,"extension":10709,"meta":10710,"navigation":10717,"path":10718,"published":10717,"seo":10719,"slug":10720,"stem":10721,"__hash__":10722},"blog\u002Fblog\u002Fpascals-triangle-and-probability.es.md","¿Qué tienen en común el triángulo de Tartaglia y una moneda?","es",{"type":8,"value":9,"toc":10690},"minimal",[10,14,18,23,40,43,49,53,56,80,83,94,104,379,384,391,527,650,813,816,1073,1377,1824,1856,2259,2263,2273,2483,2490,2556,2617,2758,2761,2771,2775,2861,2867,2874,2928,3004,3223,3227,3233,3369,3401,3841,3844,4551,4737,4740,5012,5016,5019,5103,5187,5319,5325,5457,5460,6252,6255,6261,6264,6284,6292,6296,6303,6398,6401,6597,6629,6634,6864,6923,6928,7250,7320,7325,7831,7928,7938,8314,8550,8554,8619,8931,8934,9902,10277,10280,10546,10641,10645,10648,10666,10672,10675,10678,10681],[11,12,5],"h1",{"id":13},"qué-tienen-en-común-el-triángulo-de-tartaglia-y-una-moneda",[15,16,17],"p",{},"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.",[19,20,22],"h2",{"id":21},"un-poco-de-historia","Un poco de historia",[15,24,25,26,30,31,34,35,39],{},"En muchos países de habla hispana este patrón se conoce como ",[27,28,29],"strong",{},"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 ",[27,32,33],{},"triángulo de Pascal",", porque Blaise Pascal publicó en 1654 un tratado — el ",[36,37,38],"em",{},"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.",[15,41,42],{},"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.",[15,44,45,46],{},"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 ",[36,47,48],{},"¿de cuántas maneras puedo organizar esto?",[19,50,52],{"id":51},"cómo-construir-el-triángulo","Cómo construir el triángulo",[15,54,55],{},"La regla de construcción es elegantemente simple:",[57,58,59,67,73],"ol",{},[60,61,62,63,66],"li",{},"Empieza colocando un ",[27,64,65],{},"1"," en la cima.",[60,68,69,70,72],{},"En cada fila nueva, coloca un ",[27,71,65],{}," en cada extremo.",[60,74,75,76,79],{},"Cada número interior es la ",[27,77,78],{},"suma de los dos números"," que tiene justo encima.",[15,81,82],{},"Las primeras filas quedan así:",[84,85,91],"pre",{"className":86,"code":88,"language":89,"meta":90},[87],"language-text"," 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","text","",[92,93,88],"code",{"__ignoreMap":90},[15,95,96,97,99,100,103],{},"La primera fila (el ",[27,98,65],{}," solitario de la cima) se llama ",[27,101,102],{},"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.",[15,105,106,107,211,212,241,242,211,298,326,327,378],{},"Fíjate en la fila 4: ",[108,109,112,155],"span",{"className":110},[111],"katex",[108,113,116],{"className":114},[115],"katex-mathml",[117,118,120],"math",{"xmlns":119},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[121,122,123,150],"semantics",{},[124,125,126,129,134,137,139,142,144,146,148],"mrow",{},[127,128,65],"mn",{},[130,131,133],"mo",{"separator":132},"true",",",[127,135,136],{},"4",[130,138,133],{"separator":132},[127,140,141],{},"6",[130,143,133],{"separator":132},[127,145,136],{},[130,147,133],{"separator":132},[127,149,65],{},[151,152,154],"annotation",{"encoding":153},"application\u002Fx-tex","1, 4, 6, 4, 1",[108,156,159],{"className":157,"ariaHidden":132},[158],"katex-html",[108,160,163,168,172,176,181,184,187,190,193,196,199,202,205,208],{"className":161},[162],"base",[108,164],{"className":165,"style":167},[166],"strut","height:0.8389em;vertical-align:-0.1944em;",[108,169,65],{"className":170},[171],"mord",[108,173,133],{"className":174},[175],"mpunct",[108,177],{"className":178,"style":180},[179],"mspace","margin-right:0.1667em;",[108,182,136],{"className":183},[171],[108,185,133],{"className":186},[175],[108,188],{"className":189,"style":180},[179],[108,191,141],{"className":192},[171],[108,194,133],{"className":195},[175],[108,197],{"className":198,"style":180},[179],[108,200,136],{"className":201},[171],[108,203,133],{"className":204},[175],[108,206],{"className":207,"style":180},[179],[108,209,65],{"className":210},[171],". El ",[108,213,215,228],{"className":214},[111],[108,216,218],{"className":217},[115],[117,219,220],{"xmlns":119},[121,221,222,226],{},[124,223,224],{},[127,225,141],{},[151,227,141],{"encoding":153},[108,229,231],{"className":230,"ariaHidden":132},[158],[108,232,234,238],{"className":233},[162],[108,235],{"className":236,"style":237},[166],"height:0.6444em;",[108,239,141],{"className":240},[171]," del centro viene de sumar los dos números que tiene encima: ",[108,243,245,265],{"className":244},[111],[108,246,248],{"className":247},[115],[117,249,250],{"xmlns":119},[121,251,252,262],{},[124,253,254,257,260],{},[127,255,256],{},"3",[130,258,259],{},"+",[127,261,256],{},[151,263,264],{"encoding":153},"3 + 3",[108,266,268,289],{"className":267,"ariaHidden":132},[158],[108,269,271,275,278,282,286],{"className":270},[162],[108,272],{"className":273,"style":274},[166],"height:0.7278em;vertical-align:-0.0833em;",[108,276,256],{"className":277},[171],[108,279],{"className":280,"style":281},[179],"margin-right:0.2222em;",[108,283,259],{"className":284},[285],"mbin",[108,287],{"className":288,"style":281},[179],[108,290,292,295],{"className":291},[162],[108,293],{"className":294,"style":237},[166],[108,296,256],{"className":297},[171],[108,299,301,314],{"className":300},[111],[108,302,304],{"className":303},[115],[117,305,306],{"xmlns":119},[121,307,308,312],{},[124,309,310],{},[127,311,136],{},[151,313,136],{"encoding":153},[108,315,317],{"className":316,"ariaHidden":132},[158],[108,318,320,323],{"className":319},[162],[108,321],{"className":322,"style":237},[166],[108,324,136],{"className":325},[171]," de la izquierda viene de ",[108,328,330,348],{"className":329},[111],[108,331,333],{"className":332},[115],[117,334,335],{"xmlns":119},[121,336,337,345],{},[124,338,339,341,343],{},[127,340,65],{},[130,342,259],{},[127,344,256],{},[151,346,347],{"encoding":153},"1 + 3",[108,349,351,369],{"className":350,"ariaHidden":132},[158],[108,352,354,357,360,363,366],{"className":353},[162],[108,355],{"className":356,"style":274},[166],[108,358,65],{"className":359},[171],[108,361],{"className":362,"style":281},[179],[108,364,259],{"className":365},[285],[108,367],{"className":368,"style":281},[179],[108,370,372,375],{"className":371},[162],[108,373],{"className":374,"style":237},[166],[108,376,256],{"className":377},[171],". Siempre la misma regla.",[380,381,383],"h3",{"id":382},"los-coeficientes-binomiales","Los coeficientes binomiales",[15,385,386,387,390],{},"Cada número del triángulo tiene un nombre formal: es un ",[27,388,389],{},"coeficiente binomial",". Se escribe así:",[15,392,395],{"className":393},[394],"math-block",[108,396,398,427],{"className":397},[111],[108,399,401],{"className":400},[115],[117,402,403],{"xmlns":119},[121,404,405,424],{},[124,406,407,410,421],{},[130,408,409],{"fence":132},"(",[411,412,414,418],"mfrac",{"linethickness":413},"0px",[415,416,417],"mi",{},"n",[415,419,420],{},"k",[130,422,423],{"fence":132},")",[151,425,426],{"encoding":153},"\\binom{n}{k}",[108,428,430],{"className":429,"ariaHidden":132},[158],[108,431,433,437],{"className":432},[162],[108,434],{"className":435,"style":436},[166],"height:1.2em;vertical-align:-0.35em;",[108,438,440,451,520],{"className":439},[171],[108,441,446],{"className":442,"style":445},[443,444],"mopen","delimcenter","top:0em;",[108,447,409],{"className":448},[449,450],"delimsizing","size1",[108,452,454],{"className":453},[411],[108,455,459,511],{"className":456},[457,458],"vlist-t","vlist-t2",[108,460,463,506],{"className":461},[462],"vlist-r",[108,464,468,491],{"className":465,"style":467},[466],"vlist","height:0.7454em;",[108,469,471,476],{"style":470},"top:-2.355em;",[108,472],{"className":473,"style":475},[474],"pstrut","height:2.7em;",[108,477,483],{"className":478},[479,480,481,482],"sizing","reset-size6","size3","mtight",[108,484,486],{"className":485},[171,482],[108,487,420],{"className":488,"style":490},[171,489,482],"mathnormal","margin-right:0.0315em;",[108,492,494,497],{"style":493},"top:-3.144em;",[108,495],{"className":496,"style":475},[474],[108,498,500],{"className":499},[479,480,481,482],[108,501,503],{"className":502},[171,482],[108,504,417],{"className":505},[171,489,482],[108,507,510],{"className":508},[509],"vlist-s","​",[108,512,514],{"className":513},[462],[108,515,518],{"className":516,"style":517},[466],"height:0.345em;",[108,519],{},[108,521,524],{"className":522,"style":445},[523,444],"mclose",[108,525,423],{"className":526},[449,450],[15,528,529,530,591,592,620,621,649],{},"y se lee \"",[27,531,532,561,562],{},[108,533,535,548],{"className":534},[111],[108,536,538],{"className":537},[115],[117,539,540],{"xmlns":119},[121,541,542,546],{},[124,543,544],{},[415,545,417],{},[151,547,417],{"encoding":153},[108,549,551],{"className":550,"ariaHidden":132},[158],[108,552,554,558],{"className":553},[162],[108,555],{"className":556,"style":557},[166],"height:0.4306em;",[108,559,417],{"className":560},[171,489]," sobre ",[108,563,565,578],{"className":564},[111],[108,566,568],{"className":567},[115],[117,569,570],{"xmlns":119},[121,571,572,576],{},[124,573,574],{},[415,575,420],{},[151,577,420],{"encoding":153},[108,579,581],{"className":580,"ariaHidden":132},[158],[108,582,584,588],{"className":583},[162],[108,585],{"className":586,"style":587},[166],"height:0.6944em;",[108,589,420],{"className":590,"style":490},[171,489],"\". Su significado es directo: representa el número de formas de elegir ",[108,593,595,608],{"className":594},[111],[108,596,598],{"className":597},[115],[117,599,600],{"xmlns":119},[121,601,602,606],{},[124,603,604],{},[415,605,420],{},[151,607,420],{"encoding":153},[108,609,611],{"className":610,"ariaHidden":132},[158],[108,612,614,617],{"className":613},[162],[108,615],{"className":616,"style":587},[166],[108,618,420],{"className":619,"style":490},[171,489]," elementos de un conjunto de ",[108,622,624,637],{"className":623},[111],[108,625,627],{"className":626},[115],[117,628,629],{"xmlns":119},[121,630,631,635],{},[124,632,633],{},[415,634,417],{},[151,636,417],{"encoding":153},[108,638,640],{"className":639,"ariaHidden":132},[158],[108,641,643,646],{"className":642},[162],[108,644],{"className":645,"style":557},[166],[108,647,417],{"className":648},[171,489],", sin importar el orden.",[15,651,652,653,783,784,812],{},"Por ejemplo, ",[108,654,656,686],{"className":655},[111],[108,657,659],{"className":658},[115],[117,660,661],{"xmlns":119},[121,662,663,683],{},[124,664,665,678,681],{},[124,666,667,669,676],{},[130,668,409],{"fence":132},[411,670,671,673],{"linethickness":413},[127,672,136],{},[127,674,675],{},"2",[130,677,423],{"fence":132},[130,679,680],{},"=",[127,682,141],{},[151,684,685],{"encoding":153},"\\binom{4}{2} = 6",[108,687,689,774],{"className":688,"ariaHidden":132},[158],[108,690,692,696,763,767,771],{"className":691},[162],[108,693],{"className":694,"style":695},[166],"height:1.2451em;vertical-align:-0.35em;",[108,697,699,705,757],{"className":698},[171],[108,700,702],{"className":701,"style":445},[443,444],[108,703,409],{"className":704},[449,450],[108,706,708],{"className":707},[411],[108,709,711,749],{"className":710},[457,458],[108,712,714,746],{"className":713},[462],[108,715,718,732],{"className":716,"style":717},[466],"height:0.8951em;",[108,719,720,723],{"style":470},[108,721],{"className":722,"style":475},[474],[108,724,726],{"className":725},[479,480,481,482],[108,727,729],{"className":728},[171,482],[108,730,675],{"className":731},[171,482],[108,733,734,737],{"style":493},[108,735],{"className":736,"style":475},[474],[108,738,740],{"className":739},[479,480,481,482],[108,741,743],{"className":742},[171,482],[108,744,136],{"className":745},[171,482],[108,747,510],{"className":748},[509],[108,750,752],{"className":751},[462],[108,753,755],{"className":754,"style":517},[466],[108,756],{},[108,758,760],{"className":759,"style":445},[523,444],[108,761,423],{"className":762},[449,450],[108,764],{"className":765,"style":766},[179],"margin-right:0.2778em;",[108,768,680],{"className":769},[770],"mrel",[108,772],{"className":773,"style":766},[179],[108,775,777,780],{"className":776},[162],[108,778],{"className":779,"style":237},[166],[108,781,141],{"className":782},[171]," 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 ",[108,785,787,800],{"className":786},[111],[108,788,790],{"className":789},[115],[117,791,792],{"xmlns":119},[121,793,794,798],{},[124,795,796],{},[127,797,141],{},[151,799,141],{"encoding":153},[108,801,803],{"className":802,"ariaHidden":132},[158],[108,804,806,809],{"className":805},[162],[108,807],{"className":808,"style":237},[166],[108,810,141],{"className":811},[171],".",[15,814,815],{},"La fórmula para calcularlo es:",[15,817,819],{"className":818},[394],[108,820,822,880],{"className":821},[111],[108,823,825],{"className":824},[115],[117,826,827],{"xmlns":119},[121,828,829,877],{},[124,830,831,843,845],{},[124,832,833,835,841],{},[130,834,409],{"fence":132},[411,836,837,839],{"linethickness":413},[415,838,417],{},[415,840,420],{},[130,842,423],{"fence":132},[130,844,680],{},[411,846,847,855],{},[124,848,849,851],{},[415,850,417],{},[130,852,854],{"stretchy":853},"false","!",[124,856,857,859,861,864,866,868,871,873,875],{},[415,858,420],{},[130,860,854],{"stretchy":853},[130,862,863],{},"⋅",[130,865,409],{"stretchy":853},[415,867,417],{},[130,869,870],{},"−",[415,872,420],{},[130,874,423],{"stretchy":853},[130,876,854],{"stretchy":853},[151,878,879],{"encoding":153},"\\binom{n}{k} = \\frac{n!}{k! \\cdot (n-k)!}",[108,881,883,964],{"className":882,"ariaHidden":132},[158],[108,884,886,889,955,958,961],{"className":885},[162],[108,887],{"className":888,"style":436},[166],[108,890,892,898,949],{"className":891},[171],[108,893,895],{"className":894,"style":445},[443,444],[108,896,409],{"className":897},[449,450],[108,899,901],{"className":900},[411],[108,902,904,941],{"className":903},[457,458],[108,905,907,938],{"className":906},[462],[108,908,910,924],{"className":909,"style":467},[466],[108,911,912,915],{"style":470},[108,913],{"className":914,"style":475},[474],[108,916,918],{"className":917},[479,480,481,482],[108,919,921],{"className":920},[171,482],[108,922,420],{"className":923,"style":490},[171,489,482],[108,925,926,929],{"style":493},[108,927],{"className":928,"style":475},[474],[108,930,932],{"className":931},[479,480,481,482],[108,933,935],{"className":934},[171,482],[108,936,417],{"className":937},[171,489,482],[108,939,510],{"className":940},[509],[108,942,944],{"className":943},[462],[108,945,947],{"className":946,"style":517},[466],[108,948],{},[108,950,952],{"className":951,"style":445},[523,444],[108,953,423],{"className":954},[449,450],[108,956],{"className":957,"style":766},[179],[108,959,680],{"className":960},[770],[108,962],{"className":963,"style":766},[179],[108,965,967,971],{"className":966},[162],[108,968],{"className":969,"style":970},[166],"height:1.4001em;vertical-align:-0.52em;",[108,972,974,978,1070],{"className":973},[171],[108,975],{"className":976},[443,977],"nulldelimiter",[108,979,981],{"className":980},[411],[108,982,984,1061],{"className":983},[457,458],[108,985,987,1058],{"className":986},[462],[108,988,991,1029,1040],{"className":989,"style":990},[466],"height:0.8801em;",[108,992,994,998],{"style":993},"top:-2.655em;",[108,995],{"className":996,"style":997},[474],"height:3em;",[108,999,1001],{"className":1000},[479,480,481,482],[108,1002,1004,1007,1010,1013,1016,1019,1022,1025],{"className":1003},[171,482],[108,1005,420],{"className":1006,"style":490},[171,489,482],[108,1008,854],{"className":1009},[523,482],[108,1011,863],{"className":1012},[285,482],[108,1014,409],{"className":1015},[443,482],[108,1017,417],{"className":1018},[171,489,482],[108,1020,870],{"className":1021},[285,482],[108,1023,420],{"className":1024,"style":490},[171,489,482],[108,1026,1028],{"className":1027},[523,482],")!",[108,1030,1032,1035],{"style":1031},"top:-3.23em;",[108,1033],{"className":1034,"style":997},[474],[108,1036],{"className":1037,"style":1039},[1038],"frac-line","border-bottom-width:0.04em;",[108,1041,1043,1046],{"style":1042},"top:-3.394em;",[108,1044],{"className":1045,"style":997},[474],[108,1047,1049],{"className":1048},[479,480,481,482],[108,1050,1052,1055],{"className":1051},[171,482],[108,1053,417],{"className":1054},[171,489,482],[108,1056,854],{"className":1057},[523,482],[108,1059,510],{"className":1060},[509],[108,1062,1064],{"className":1063},[462],[108,1065,1068],{"className":1066,"style":1067},[466],"height:0.52em;",[108,1069],{},[108,1071],{"className":1072},[523,977],[15,1074,1075,1076,1110,1111,1114,1115,1143,1144,1172,1173,1319,1320,812],{},"Aquí, el símbolo ",[108,1077,1079,1095],{"className":1078},[111],[108,1080,1082],{"className":1081},[115],[117,1083,1084],{"xmlns":119},[121,1085,1086,1092],{},[124,1087,1088,1090],{},[415,1089,417],{},[130,1091,854],{"stretchy":853},[151,1093,1094],{"encoding":153},"n!",[108,1096,1098],{"className":1097,"ariaHidden":132},[158],[108,1099,1101,1104,1107],{"className":1100},[162],[108,1102],{"className":1103,"style":587},[166],[108,1105,417],{"className":1106},[171,489],[108,1108,854],{"className":1109},[523]," se llama ",[27,1112,1113],{},"factorial"," y significa multiplicar todos los números enteros desde ",[108,1116,1118,1131],{"className":1117},[111],[108,1119,1121],{"className":1120},[115],[117,1122,1123],{"xmlns":119},[121,1124,1125,1129],{},[124,1126,1127],{},[127,1128,65],{},[151,1130,65],{"encoding":153},[108,1132,1134],{"className":1133,"ariaHidden":132},[158],[108,1135,1137,1140],{"className":1136},[162],[108,1138],{"className":1139,"style":237},[166],[108,1141,65],{"className":1142},[171]," hasta ",[108,1145,1147,1160],{"className":1146},[111],[108,1148,1150],{"className":1149},[115],[117,1151,1152],{"xmlns":119},[121,1153,1154,1158],{},[124,1155,1156],{},[415,1157,417],{},[151,1159,417],{"encoding":153},[108,1161,1163],{"className":1162,"ariaHidden":132},[158],[108,1164,1166,1169],{"className":1165},[162],[108,1167],{"className":1168,"style":557},[166],[108,1170,417],{"className":1171},[171,489],". Por ejemplo, ",[108,1174,1176,1214],{"className":1175},[111],[108,1177,1179],{"className":1178},[115],[117,1180,1181],{"xmlns":119},[121,1182,1183,1211],{},[124,1184,1185,1187,1189,1191,1193,1196,1198,1200,1202,1204,1206,1208],{},[127,1186,136],{},[130,1188,854],{"stretchy":853},[130,1190,680],{},[127,1192,136],{},[130,1194,1195],{},"×",[127,1197,256],{},[130,1199,1195],{},[127,1201,675],{},[130,1203,1195],{},[127,1205,65],{},[130,1207,680],{},[127,1209,1210],{},"24",[151,1212,1213],{"encoding":153},"4! = 4 \\times 3 \\times 2 \\times 1 = 24",[108,1215,1217,1238,1256,1274,1292,1310],{"className":1216,"ariaHidden":132},[158],[108,1218,1220,1223,1226,1229,1232,1235],{"className":1219},[162],[108,1221],{"className":1222,"style":587},[166],[108,1224,136],{"className":1225},[171],[108,1227,854],{"className":1228},[523],[108,1230],{"className":1231,"style":766},[179],[108,1233,680],{"className":1234},[770],[108,1236],{"className":1237,"style":766},[179],[108,1239,1241,1244,1247,1250,1253],{"className":1240},[162],[108,1242],{"className":1243,"style":274},[166],[108,1245,136],{"className":1246},[171],[108,1248],{"className":1249,"style":281},[179],[108,1251,1195],{"className":1252},[285],[108,1254],{"className":1255,"style":281},[179],[108,1257,1259,1262,1265,1268,1271],{"className":1258},[162],[108,1260],{"className":1261,"style":274},[166],[108,1263,256],{"className":1264},[171],[108,1266],{"className":1267,"style":281},[179],[108,1269,1195],{"className":1270},[285],[108,1272],{"className":1273,"style":281},[179],[108,1275,1277,1280,1283,1286,1289],{"className":1276},[162],[108,1278],{"className":1279,"style":274},[166],[108,1281,675],{"className":1282},[171],[108,1284],{"className":1285,"style":281},[179],[108,1287,1195],{"className":1288},[285],[108,1290],{"className":1291,"style":281},[179],[108,1293,1295,1298,1301,1304,1307],{"className":1294},[162],[108,1296],{"className":1297,"style":237},[166],[108,1299,65],{"className":1300},[171],[108,1302],{"className":1303,"style":766},[179],[108,1305,680],{"className":1306},[770],[108,1308],{"className":1309,"style":766},[179],[108,1311,1313,1316],{"className":1312},[162],[108,1314],{"className":1315,"style":237},[166],[108,1317,1210],{"className":1318},[171],". Por convención, ",[108,1321,1323,1344],{"className":1322},[111],[108,1324,1326],{"className":1325},[115],[117,1327,1328],{"xmlns":119},[121,1329,1330,1341],{},[124,1331,1332,1335,1337,1339],{},[127,1333,1334],{},"0",[130,1336,854],{"stretchy":853},[130,1338,680],{},[127,1340,65],{},[151,1342,1343],{"encoding":153},"0! = 1",[108,1345,1347,1368],{"className":1346,"ariaHidden":132},[158],[108,1348,1350,1353,1356,1359,1362,1365],{"className":1349},[162],[108,1351],{"className":1352,"style":587},[166],[108,1354,1334],{"className":1355},[171],[108,1357,854],{"className":1358},[523],[108,1360],{"className":1361,"style":766},[179],[108,1363,680],{"className":1364},[770],[108,1366],{"className":1367,"style":766},[179],[108,1369,1371,1374],{"className":1370},[162],[108,1372],{"className":1373,"style":237},[166],[108,1375,65],{"className":1376},[171],[15,1378,1379,1380,1823],{},"Comprobemos: ",[108,1381,1383,1455],{"className":1382},[111],[108,1384,1386],{"className":1385},[115],[117,1387,1388],{"xmlns":119},[121,1389,1390,1452],{},[124,1391,1392,1404,1406,1426,1428,1440,1442,1448,1450],{},[124,1393,1394,1396,1402],{},[130,1395,409],{"fence":132},[411,1397,1398,1400],{"linethickness":413},[127,1399,136],{},[127,1401,675],{},[130,1403,423],{"fence":132},[130,1405,680],{},[411,1407,1408,1414],{},[124,1409,1410,1412],{},[127,1411,136],{},[130,1413,854],{"stretchy":853},[124,1415,1416,1418,1420,1422,1424],{},[127,1417,675],{},[130,1419,854],{"stretchy":853},[130,1421,863],{},[127,1423,675],{},[130,1425,854],{"stretchy":853},[130,1427,680],{},[411,1429,1430,1432],{},[127,1431,1210],{},[124,1433,1434,1436,1438],{},[127,1435,675],{},[130,1437,1195],{},[127,1439,675],{},[130,1441,680],{},[411,1443,1444,1446],{},[127,1445,1210],{},[127,1447,136],{},[130,1449,680],{},[127,1451,141],{},[151,1453,1454],{"encoding":153},"\\binom{4}{2} = \\frac{4!}{2! \\cdot 2!} = \\frac{24}{2 \\times 2} = \\frac{24}{4} = 6",[108,1456,1458,1539,1638,1730,1814],{"className":1457,"ariaHidden":132},[158],[108,1459,1461,1464,1530,1533,1536],{"className":1460},[162],[108,1462],{"className":1463,"style":695},[166],[108,1465,1467,1473,1524],{"className":1466},[171],[108,1468,1470],{"className":1469,"style":445},[443,444],[108,1471,409],{"className":1472},[449,450],[108,1474,1476],{"className":1475},[411],[108,1477,1479,1516],{"className":1478},[457,458],[108,1480,1482,1513],{"className":1481},[462],[108,1483,1485,1499],{"className":1484,"style":717},[466],[108,1486,1487,1490],{"style":470},[108,1488],{"className":1489,"style":475},[474],[108,1491,1493],{"className":1492},[479,480,481,482],[108,1494,1496],{"className":1495},[171,482],[108,1497,675],{"className":1498},[171,482],[108,1500,1501,1504],{"style":493},[108,1502],{"className":1503,"style":475},[474],[108,1505,1507],{"className":1506},[479,480,481,482],[108,1508,1510],{"className":1509},[171,482],[108,1511,136],{"className":1512},[171,482],[108,1514,510],{"className":1515},[509],[108,1517,1519],{"className":1518},[462],[108,1520,1522],{"className":1521,"style":517},[466],[108,1523],{},[108,1525,1527],{"className":1526,"style":445},[523,444],[108,1528,423],{"className":1529},[449,450],[108,1531],{"className":1532,"style":766},[179],[108,1534,680],{"className":1535},[770],[108,1537],{"className":1538,"style":766},[179],[108,1540,1542,1546,1629,1632,1635],{"className":1541},[162],[108,1543],{"className":1544,"style":1545},[166],"height:1.2251em;vertical-align:-0.345em;",[108,1547,1549,1552,1626],{"className":1548},[171],[108,1550],{"className":1551},[443,977],[108,1553,1555],{"className":1554},[411],[108,1556,1558,1618],{"className":1557},[457,458],[108,1559,1561,1615],{"className":1560},[462],[108,1562,1564,1590,1598],{"className":1563,"style":990},[466],[108,1565,1566,1569],{"style":993},[108,1567],{"className":1568,"style":997},[474],[108,1570,1572],{"className":1571},[479,480,481,482],[108,1573,1575,1578,1581,1584,1587],{"className":1574},[171,482],[108,1576,675],{"className":1577},[171,482],[108,1579,854],{"className":1580},[523,482],[108,1582,863],{"className":1583},[285,482],[108,1585,675],{"className":1586},[171,482],[108,1588,854],{"className":1589},[523,482],[108,1591,1592,1595],{"style":1031},[108,1593],{"className":1594,"style":997},[474],[108,1596],{"className":1597,"style":1039},[1038],[108,1599,1600,1603],{"style":1042},[108,1601],{"className":1602,"style":997},[474],[108,1604,1606],{"className":1605},[479,480,481,482],[108,1607,1609,1612],{"className":1608},[171,482],[108,1610,136],{"className":1611},[171,482],[108,1613,854],{"className":1614},[523,482],[108,1616,510],{"className":1617},[509],[108,1619,1621],{"className":1620},[462],[108,1622,1624],{"className":1623,"style":517},[466],[108,1625],{},[108,1627],{"className":1628},[523,977],[108,1630],{"className":1631,"style":766},[179],[108,1633,680],{"className":1634},[770],[108,1636],{"className":1637,"style":766},[179],[108,1639,1641,1645,1721,1724,1727],{"className":1640},[162],[108,1642],{"className":1643,"style":1644},[166],"height:1.2484em;vertical-align:-0.4033em;",[108,1646,1648,1651,1718],{"className":1647},[171],[108,1649],{"className":1650},[443,977],[108,1652,1654],{"className":1653},[411],[108,1655,1657,1709],{"className":1656},[457,458],[108,1658,1660,1706],{"className":1659},[462],[108,1661,1664,1684,1692],{"className":1662,"style":1663},[466],"height:0.8451em;",[108,1665,1666,1669],{"style":993},[108,1667],{"className":1668,"style":997},[474],[108,1670,1672],{"className":1671},[479,480,481,482],[108,1673,1675,1678,1681],{"className":1674},[171,482],[108,1676,675],{"className":1677},[171,482],[108,1679,1195],{"className":1680},[285,482],[108,1682,675],{"className":1683},[171,482],[108,1685,1686,1689],{"style":1031},[108,1687],{"className":1688,"style":997},[474],[108,1690],{"className":1691,"style":1039},[1038],[108,1693,1694,1697],{"style":1042},[108,1695],{"className":1696,"style":997},[474],[108,1698,1700],{"className":1699},[479,480,481,482],[108,1701,1703],{"className":1702},[171,482],[108,1704,1210],{"className":1705},[171,482],[108,1707,510],{"className":1708},[509],[108,1710,1712],{"className":1711},[462],[108,1713,1716],{"className":1714,"style":1715},[466],"height:0.4033em;",[108,1717],{},[108,1719],{"className":1720},[523,977],[108,1722],{"className":1723,"style":766},[179],[108,1725,680],{"className":1726},[770],[108,1728],{"className":1729,"style":766},[179],[108,1731,1733,1737,1805,1808,1811],{"className":1732},[162],[108,1734],{"className":1735,"style":1736},[166],"height:1.1901em;vertical-align:-0.345em;",[108,1738,1740,1743,1802],{"className":1739},[171],[108,1741],{"className":1742},[443,977],[108,1744,1746],{"className":1745},[411],[108,1747,1749,1794],{"className":1748},[457,458],[108,1750,1752,1791],{"className":1751},[462],[108,1753,1755,1769,1777],{"className":1754,"style":1663},[466],[108,1756,1757,1760],{"style":993},[108,1758],{"className":1759,"style":997},[474],[108,1761,1763],{"className":1762},[479,480,481,482],[108,1764,1766],{"className":1765},[171,482],[108,1767,136],{"className":1768},[171,482],[108,1770,1771,1774],{"style":1031},[108,1772],{"className":1773,"style":997},[474],[108,1775],{"className":1776,"style":1039},[1038],[108,1778,1779,1782],{"style":1042},[108,1780],{"className":1781,"style":997},[474],[108,1783,1785],{"className":1784},[479,480,481,482],[108,1786,1788],{"className":1787},[171,482],[108,1789,1210],{"className":1790},[171,482],[108,1792,510],{"className":1793},[509],[108,1795,1797],{"className":1796},[462],[108,1798,1800],{"className":1799,"style":517},[466],[108,1801],{},[108,1803],{"className":1804},[523,977],[108,1806],{"className":1807,"style":766},[179],[108,1809,680],{"className":1810},[770],[108,1812],{"className":1813,"style":766},[179],[108,1815,1817,1820],{"className":1816},[162],[108,1818],{"className":1819,"style":237},[166],[108,1821,141],{"className":1822},[171],". Correcto.",[15,1825,1826,1827,1855],{},"Así, la fila ",[108,1828,1830,1843],{"className":1829},[111],[108,1831,1833],{"className":1832},[115],[117,1834,1835],{"xmlns":119},[121,1836,1837,1841],{},[124,1838,1839],{},[415,1840,417],{},[151,1842,417],{"encoding":153},[108,1844,1846],{"className":1845,"ariaHidden":132},[158],[108,1847,1849,1852],{"className":1848},[162],[108,1850],{"className":1851,"style":557},[166],[108,1853,417],{"className":1854},[171,489]," del triángulo contiene los valores:",[15,1857,1859],{"className":1858},[394],[108,1860,1862,1943],{"className":1861},[111],[108,1863,1865],{"className":1864},[115],[117,1866,1867],{"xmlns":119},[121,1868,1869,1940],{},[124,1870,1871,1883,1885,1889,1901,1903,1905,1917,1919,1921,1924,1926,1928],{},[124,1872,1873,1875,1881],{},[130,1874,409],{"fence":132},[411,1876,1877,1879],{"linethickness":413},[415,1878,417],{},[127,1880,1334],{},[130,1882,423],{"fence":132},[130,1884,133],{"separator":132},[1886,1887,1888],"mtext",{},"  ",[124,1890,1891,1893,1899],{},[130,1892,409],{"fence":132},[411,1894,1895,1897],{"linethickness":413},[415,1896,417],{},[127,1898,65],{},[130,1900,423],{"fence":132},[130,1902,133],{"separator":132},[1886,1904,1888],{},[124,1906,1907,1909,1915],{},[130,1908,409],{"fence":132},[411,1910,1911,1913],{"linethickness":413},[415,1912,417],{},[127,1914,675],{},[130,1916,423],{"fence":132},[130,1918,133],{"separator":132},[1886,1920,1888],{},[130,1922,1923],{},"…",[130,1925,133],{"separator":132},[1886,1927,1888],{},[124,1929,1930,1932,1938],{},[130,1931,409],{"fence":132},[411,1933,1934,1936],{"linethickness":413},[415,1935,417],{},[415,1937,417],{},[130,1939,423],{"fence":132},[151,1941,1942],{"encoding":153},"\\binom{n}{0},\\; \\binom{n}{1},\\; \\binom{n}{2},\\; \\ldots,\\; \\binom{n}{n}",[108,1944,1946],{"className":1945,"ariaHidden":132},[158],[108,1947,1949,1952,2018,2021,2024,2027,2093,2096,2099,2102,2168,2171,2174,2177,2181,2184,2187,2190,2193],{"className":1948},[162],[108,1950],{"className":1951,"style":436},[166],[108,1953,1955,1961,2012],{"className":1954},[171],[108,1956,1958],{"className":1957,"style":445},[443,444],[108,1959,409],{"className":1960},[449,450],[108,1962,1964],{"className":1963},[411],[108,1965,1967,2004],{"className":1966},[457,458],[108,1968,1970,2001],{"className":1969},[462],[108,1971,1973,1987],{"className":1972,"style":467},[466],[108,1974,1975,1978],{"style":470},[108,1976],{"className":1977,"style":475},[474],[108,1979,1981],{"className":1980},[479,480,481,482],[108,1982,1984],{"className":1983},[171,482],[108,1985,1334],{"className":1986},[171,482],[108,1988,1989,1992],{"style":493},[108,1990],{"className":1991,"style":475},[474],[108,1993,1995],{"className":1994},[479,480,481,482],[108,1996,1998],{"className":1997},[171,482],[108,1999,417],{"className":2000},[171,489,482],[108,2002,510],{"className":2003},[509],[108,2005,2007],{"className":2006},[462],[108,2008,2010],{"className":2009,"style":517},[466],[108,2011],{},[108,2013,2015],{"className":2014,"style":445},[523,444],[108,2016,423],{"className":2017},[449,450],[108,2019,133],{"className":2020},[175],[108,2022],{"className":2023,"style":766},[179],[108,2025],{"className":2026,"style":180},[179],[108,2028,2030,2036,2087],{"className":2029},[171],[108,2031,2033],{"className":2032,"style":445},[443,444],[108,2034,409],{"className":2035},[449,450],[108,2037,2039],{"className":2038},[411],[108,2040,2042,2079],{"className":2041},[457,458],[108,2043,2045,2076],{"className":2044},[462],[108,2046,2048,2062],{"className":2047,"style":467},[466],[108,2049,2050,2053],{"style":470},[108,2051],{"className":2052,"style":475},[474],[108,2054,2056],{"className":2055},[479,480,481,482],[108,2057,2059],{"className":2058},[171,482],[108,2060,65],{"className":2061},[171,482],[108,2063,2064,2067],{"style":493},[108,2065],{"className":2066,"style":475},[474],[108,2068,2070],{"className":2069},[479,480,481,482],[108,2071,2073],{"className":2072},[171,482],[108,2074,417],{"className":2075},[171,489,482],[108,2077,510],{"className":2078},[509],[108,2080,2082],{"className":2081},[462],[108,2083,2085],{"className":2084,"style":517},[466],[108,2086],{},[108,2088,2090],{"className":2089,"style":445},[523,444],[108,2091,423],{"className":2092},[449,450],[108,2094,133],{"className":2095},[175],[108,2097],{"className":2098,"style":766},[179],[108,2100],{"className":2101,"style":180},[179],[108,2103,2105,2111,2162],{"className":2104},[171],[108,2106,2108],{"className":2107,"style":445},[443,444],[108,2109,409],{"className":2110},[449,450],[108,2112,2114],{"className":2113},[411],[108,2115,2117,2154],{"className":2116},[457,458],[108,2118,2120,2151],{"className":2119},[462],[108,2121,2123,2137],{"className":2122,"style":467},[466],[108,2124,2125,2128],{"style":470},[108,2126],{"className":2127,"style":475},[474],[108,2129,2131],{"className":2130},[479,480,481,482],[108,2132,2134],{"className":2133},[171,482],[108,2135,675],{"className":2136},[171,482],[108,2138,2139,2142],{"style":493},[108,2140],{"className":2141,"style":475},[474],[108,2143,2145],{"className":2144},[479,480,481,482],[108,2146,2148],{"className":2147},[171,482],[108,2149,417],{"className":2150},[171,489,482],[108,2152,510],{"className":2153},[509],[108,2155,2157],{"className":2156},[462],[108,2158,2160],{"className":2159,"style":517},[466],[108,2161],{},[108,2163,2165],{"className":2164,"style":445},[523,444],[108,2166,423],{"className":2167},[449,450],[108,2169,133],{"className":2170},[175],[108,2172],{"className":2173,"style":766},[179],[108,2175],{"className":2176,"style":180},[179],[108,2178,1923],{"className":2179},[2180],"minner",[108,2182],{"className":2183,"style":180},[179],[108,2185,133],{"className":2186},[175],[108,2188],{"className":2189,"style":766},[179],[108,2191],{"className":2192,"style":180},[179],[108,2194,2196,2202,2253],{"className":2195},[171],[108,2197,2199],{"className":2198,"style":445},[443,444],[108,2200,409],{"className":2201},[449,450],[108,2203,2205],{"className":2204},[411],[108,2206,2208,2245],{"className":2207},[457,458],[108,2209,2211,2242],{"className":2210},[462],[108,2212,2214,2228],{"className":2213,"style":467},[466],[108,2215,2216,2219],{"style":470},[108,2217],{"className":2218,"style":475},[474],[108,2220,2222],{"className":2221},[479,480,481,482],[108,2223,2225],{"className":2224},[171,482],[108,2226,417],{"className":2227},[171,489,482],[108,2229,2230,2233],{"style":493},[108,2231],{"className":2232,"style":475},[474],[108,2234,2236],{"className":2235},[479,480,481,482],[108,2237,2239],{"className":2238},[171,482],[108,2240,417],{"className":2241},[171,489,482],[108,2243,510],{"className":2244},[509],[108,2246,2248],{"className":2247},[462],[108,2249,2251],{"className":2250,"style":517},[466],[108,2252],{},[108,2254,2256],{"className":2255,"style":445},[523,444],[108,2257,423],{"className":2258},[449,450],[19,2260,2262],{"id":2261},"la-combinatoria-el-arte-de-contar-sin-enumerar","La combinatoria: el arte de contar sin enumerar",[15,2264,2265,2266,2269,2270,812],{},"La ",[27,2267,2268],{},"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 ",[36,2271,2272],{},"sin tener que listarlas una por una",[15,2274,2275,2276,2279,2280,2283,2284,2312,2313,2386,2387,812],{},"Para entender por qué esto es útil, pensemos en lanzar una moneda. Cada lanzamiento tiene dos resultados posibles: ",[27,2277,2278],{},"cara"," (O) o ",[27,2281,2282],{},"cruz"," (X). Si lanzamos la moneda una vez, hay ",[108,2285,2287,2300],{"className":2286},[111],[108,2288,2290],{"className":2289},[115],[117,2291,2292],{"xmlns":119},[121,2293,2294,2298],{},[124,2295,2296],{},[127,2297,675],{},[151,2299,675],{"encoding":153},[108,2301,2303],{"className":2302,"ariaHidden":132},[158],[108,2304,2306,2309],{"className":2305},[162],[108,2307],{"className":2308,"style":237},[166],[108,2310,675],{"className":2311},[171]," resultados. Si la lanzamos dos veces, cada uno de los 2 resultados del primer lanzamiento se combina con los 2 del segundo, dando ",[108,2314,2316,2338],{"className":2315},[111],[108,2317,2319],{"className":2318},[115],[117,2320,2321],{"xmlns":119},[121,2322,2323,2335],{},[124,2324,2325,2327,2329,2331,2333],{},[127,2326,675],{},[130,2328,1195],{},[127,2330,675],{},[130,2332,680],{},[127,2334,136],{},[151,2336,2337],{"encoding":153},"2 \\times 2 = 4",[108,2339,2341,2359,2377],{"className":2340,"ariaHidden":132},[158],[108,2342,2344,2347,2350,2353,2356],{"className":2343},[162],[108,2345],{"className":2346,"style":274},[166],[108,2348,675],{"className":2349},[171],[108,2351],{"className":2352,"style":281},[179],[108,2354,1195],{"className":2355},[285],[108,2357],{"className":2358,"style":281},[179],[108,2360,2362,2365,2368,2371,2374],{"className":2361},[162],[108,2363],{"className":2364,"style":237},[166],[108,2366,675],{"className":2367},[171],[108,2369],{"className":2370,"style":766},[179],[108,2372,680],{"className":2373},[770],[108,2375],{"className":2376,"style":766},[179],[108,2378,2380,2383],{"className":2379},[162],[108,2381],{"className":2382,"style":237},[166],[108,2384,136],{"className":2385},[171]," resultados. Con tres lanzamientos, ",[108,2388,2390,2417],{"className":2389},[111],[108,2391,2393],{"className":2392},[115],[117,2394,2395],{"xmlns":119},[121,2396,2397,2414],{},[124,2398,2399,2401,2403,2405,2407,2409,2411],{},[127,2400,675],{},[130,2402,1195],{},[127,2404,675],{},[130,2406,1195],{},[127,2408,675],{},[130,2410,680],{},[127,2412,2413],{},"8",[151,2415,2416],{"encoding":153},"2 \\times 2 \\times 2 = 8",[108,2418,2420,2438,2456,2474],{"className":2419,"ariaHidden":132},[158],[108,2421,2423,2426,2429,2432,2435],{"className":2422},[162],[108,2424],{"className":2425,"style":274},[166],[108,2427,675],{"className":2428},[171],[108,2430],{"className":2431,"style":281},[179],[108,2433,1195],{"className":2434},[285],[108,2436],{"className":2437,"style":281},[179],[108,2439,2441,2444,2447,2450,2453],{"className":2440},[162],[108,2442],{"className":2443,"style":274},[166],[108,2445,675],{"className":2446},[171],[108,2448],{"className":2449,"style":281},[179],[108,2451,1195],{"className":2452},[285],[108,2454],{"className":2455,"style":281},[179],[108,2457,2459,2462,2465,2468,2471],{"className":2458},[162],[108,2460],{"className":2461,"style":237},[166],[108,2463,675],{"className":2464},[171],[108,2466],{"className":2467,"style":766},[179],[108,2469,680],{"className":2470},[770],[108,2472],{"className":2473,"style":766},[179],[108,2475,2477,2480],{"className":2476},[162],[108,2478],{"className":2479,"style":237},[166],[108,2481,2413],{"className":2482},[171],[15,2484,2485,2486,2489],{},"Ese patrón se expresa con un ",[27,2487,2488],{},"exponente",":",[15,2491,2493],{"className":2492},[394],[108,2494,2496,2515],{"className":2495},[111],[108,2497,2499],{"className":2498},[115],[117,2500,2501],{"xmlns":119},[121,2502,2503,2512],{},[124,2504,2505],{},[2506,2507,2508,2510],"msup",{},[127,2509,675],{},[415,2511,417],{},[151,2513,2514],{"encoding":153},"2^n",[108,2516,2518],{"className":2517,"ariaHidden":132},[158],[108,2519,2521,2525],{"className":2520},[162],[108,2522],{"className":2523,"style":2524},[166],"height:0.6644em;",[108,2526,2528,2531],{"className":2527},[171],[108,2529,675],{"className":2530},[171],[108,2532,2535],{"className":2533},[2534],"msupsub",[108,2536,2538],{"className":2537},[457],[108,2539,2541],{"className":2540},[462],[108,2542,2544],{"className":2543,"style":2524},[466],[108,2545,2547,2550],{"style":2546},"top:-3.063em;margin-right:0.05em;",[108,2548],{"className":2549,"style":475},[474],[108,2551,2553],{"className":2552},[479,480,481,482],[108,2554,417],{"className":2555},[171,489,482],[15,2557,2558,2559,2587,2588,2616],{},"donde ",[108,2560,2562,2575],{"className":2561},[111],[108,2563,2565],{"className":2564},[115],[117,2566,2567],{"xmlns":119},[121,2568,2569,2573],{},[124,2570,2571],{},[415,2572,417],{},[151,2574,417],{"encoding":153},[108,2576,2578],{"className":2577,"ariaHidden":132},[158],[108,2579,2581,2584],{"className":2580},[162],[108,2582],{"className":2583,"style":557},[166],[108,2585,417],{"className":2586},[171,489]," es el número de lanzamientos. El exponente indica cuántas veces multiplicamos la base (",[108,2589,2591,2604],{"className":2590},[111],[108,2592,2594],{"className":2593},[115],[117,2595,2596],{"xmlns":119},[121,2597,2598,2602],{},[124,2599,2600],{},[127,2601,675],{},[151,2603,675],{"encoding":153},[108,2605,2607],{"className":2606,"ariaHidden":132},[158],[108,2608,2610,2613],{"className":2609},[162],[108,2611],{"className":2612,"style":237},[166],[108,2614,675],{"className":2615},[171],") por sí misma. Así:",[2618,2619,2620,2720],"table",{},[2621,2622,2623],"thead",{},[2624,2625,2626,2659],"tr",{},[2627,2628,2630,2631,423],"th",{"align":2629},"center","Lanzamientos (",[108,2632,2634,2647],{"className":2633},[111],[108,2635,2637],{"className":2636},[115],[117,2638,2639],{"xmlns":119},[121,2640,2641,2645],{},[124,2642,2643],{},[415,2644,417],{},[151,2646,417],{"encoding":153},[108,2648,2650],{"className":2649,"ariaHidden":132},[158],[108,2651,2653,2656],{"className":2652},[162],[108,2654],{"className":2655,"style":557},[166],[108,2657,417],{"className":2658},[171,489],[2627,2660,2661,2662,423],{"align":2629},"Resultados posibles (",[108,2663,2665,2682],{"className":2664},[111],[108,2666,2668],{"className":2667},[115],[117,2669,2670],{"xmlns":119},[121,2671,2672,2680],{},[124,2673,2674],{},[2506,2675,2676,2678],{},[127,2677,675],{},[415,2679,417],{},[151,2681,2514],{"encoding":153},[108,2683,2685],{"className":2684,"ariaHidden":132},[158],[108,2686,2688,2691],{"className":2687},[162],[108,2689],{"className":2690,"style":2524},[166],[108,2692,2694,2697],{"className":2693},[171],[108,2695,675],{"className":2696},[171],[108,2698,2700],{"className":2699},[2534],[108,2701,2703],{"className":2702},[457],[108,2704,2706],{"className":2705},[462],[108,2707,2709],{"className":2708,"style":2524},[466],[108,2710,2711,2714],{"style":2546},[108,2712],{"className":2713,"style":475},[474],[108,2715,2717],{"className":2716},[479,480,481,482],[108,2718,417],{"className":2719},[171,489,482],[2721,2722,2723,2730,2736,2742,2750],"tbody",{},[2624,2724,2725,2728],{},[2726,2727,65],"td",{"align":2629},[2726,2729,675],{"align":2629},[2624,2731,2732,2734],{},[2726,2733,675],{"align":2629},[2726,2735,136],{"align":2629},[2624,2737,2738,2740],{},[2726,2739,256],{"align":2629},[2726,2741,2413],{"align":2629},[2624,2743,2744,2747],{},[2726,2745,2746],{"align":2629},"5",[2726,2748,2749],{"align":2629},"32",[2624,2751,2752,2755],{},[2726,2753,2754],{"align":2629},"10",[2726,2756,2757],{"align":2629},"1.024",[15,2759,2760],{},"Con solo 10 lanzamientos ya hay más de mil combinaciones posibles. La combinatoria nos ayuda a manejar esas cantidades sin perder la cabeza.",[15,2762,2763,2764,812],{},"Si necesitas repasar qué es la probabilidad y cómo se mide, echa un vistazo a ",[2765,2766,2770],"a",{"href":2767,"rel":2768},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Ffundamentos-de-probabilidad\u002F",[2769],"nofollow","Fundamentos de probabilidad",[19,2772,2774],{"id":2773},"caso-práctico-3-lanzamientos","Caso práctico: 3 lanzamientos",[15,2776,2777,2778,2860],{},"Vamos a conectar todo con un ejemplo concreto. Lanzamos una moneda 3 veces. Sabemos que hay ",[108,2779,2781,2803],{"className":2780},[111],[108,2782,2784],{"className":2783},[115],[117,2785,2786],{"xmlns":119},[121,2787,2788,2800],{},[124,2789,2790,2796,2798],{},[2506,2791,2792,2794],{},[127,2793,675],{},[127,2795,256],{},[130,2797,680],{},[127,2799,2413],{},[151,2801,2802],{"encoding":153},"2^3 = 8",[108,2804,2806,2851],{"className":2805,"ariaHidden":132},[158],[108,2807,2809,2813,2842,2845,2848],{"className":2808},[162],[108,2810],{"className":2811,"style":2812},[166],"height:0.8141em;",[108,2814,2816,2819],{"className":2815},[171],[108,2817,675],{"className":2818},[171],[108,2820,2822],{"className":2821},[2534],[108,2823,2825],{"className":2824},[457],[108,2826,2828],{"className":2827},[462],[108,2829,2831],{"className":2830,"style":2812},[466],[108,2832,2833,2836],{"style":2546},[108,2834],{"className":2835,"style":475},[474],[108,2837,2839],{"className":2838},[479,480,481,482],[108,2840,256],{"className":2841},[171,482],[108,2843],{"className":2844,"style":766},[179],[108,2846,680],{"className":2847},[770],[108,2849],{"className":2850,"style":766},[179],[108,2852,2854,2857],{"className":2853},[162],[108,2855],{"className":2856,"style":237},[166],[108,2858,2413],{"className":2859},[171]," resultados posibles. Aquí están todos:",[84,2862,2865],{"className":2863,"code":2864,"language":89,"meta":90},[87],"OOO OOX OXO XOO OXX XOX XXO XXX\n",[92,2866,2864],{"__ignoreMap":90},[15,2868,2869,2870,2873],{},"Ahora agrupemos estos resultados según el ",[27,2871,2872],{},"número de caras"," obtenidas:",[2618,2875,2876,2890],{},[2621,2877,2878],{},[2624,2879,2880,2883,2887],{},[2627,2881,2882],{"align":2629},"Caras",[2627,2884,2886],{"align":2885},"left","Combinaciones",[2627,2888,2889],{"align":2629},"Cantidad",[2721,2891,2892,2901,2910,2919],{},[2624,2893,2894,2896,2899],{},[2726,2895,1334],{"align":2629},[2726,2897,2898],{"align":2885},"XXX",[2726,2900,65],{"align":2629},[2624,2902,2903,2905,2908],{},[2726,2904,65],{"align":2629},[2726,2906,2907],{"align":2885},"OXX, XOX, XXO",[2726,2909,256],{"align":2629},[2624,2911,2912,2914,2917],{},[2726,2913,675],{"align":2629},[2726,2915,2916],{"align":2885},"OOX, OXO, XOO",[2726,2918,256],{"align":2629},[2624,2920,2921,2923,2926],{},[2726,2922,256],{"align":2629},[2726,2924,2925],{"align":2885},"OOO",[2726,2927,65],{"align":2629},[15,2929,2930,2931,2999,3000,3003],{},"Mira la columna de cantidades: ",[108,2932,2934,2960],{"className":2933},[111],[108,2935,2937],{"className":2936},[115],[117,2938,2939],{"xmlns":119},[121,2940,2941,2957],{},[124,2942,2943,2945,2947,2949,2951,2953,2955],{},[127,2944,65],{},[130,2946,133],{"separator":132},[127,2948,256],{},[130,2950,133],{"separator":132},[127,2952,256],{},[130,2954,133],{"separator":132},[127,2956,65],{},[151,2958,2959],{"encoding":153},"1, 3, 3, 1",[108,2961,2963],{"className":2962,"ariaHidden":132},[158],[108,2964,2966,2969,2972,2975,2978,2981,2984,2987,2990,2993,2996],{"className":2965},[162],[108,2967],{"className":2968,"style":167},[166],[108,2970,65],{"className":2971},[171],[108,2973,133],{"className":2974},[175],[108,2976],{"className":2977,"style":180},[179],[108,2979,256],{"className":2980},[171],[108,2982,133],{"className":2983},[175],[108,2985],{"className":2986,"style":180},[179],[108,2988,256],{"className":2989},[171],[108,2991,133],{"className":2992},[175],[108,2994],{"className":2995,"style":180},[179],[108,2997,65],{"className":2998},[171],". ¿Te suena? Es exactamente la ",[27,3001,3002],{},"fila 3"," del triángulo de Tartaglia.",[15,3005,3006,3007,3035,3036,3064,3065,3164,3165,3193,3194,3222],{},"Y no es casualidad. El número de formas de obtener exactamente ",[108,3008,3010,3023],{"className":3009},[111],[108,3011,3013],{"className":3012},[115],[117,3014,3015],{"xmlns":119},[121,3016,3017,3021],{},[124,3018,3019],{},[415,3020,420],{},[151,3022,420],{"encoding":153},[108,3024,3026],{"className":3025,"ariaHidden":132},[158],[108,3027,3029,3032],{"className":3028},[162],[108,3030],{"className":3031,"style":587},[166],[108,3033,420],{"className":3034,"style":490},[171,489]," caras en ",[108,3037,3039,3052],{"className":3038},[111],[108,3040,3042],{"className":3041},[115],[117,3043,3044],{"xmlns":119},[121,3045,3046,3050],{},[124,3047,3048],{},[415,3049,417],{},[151,3051,417],{"encoding":153},[108,3053,3055],{"className":3054,"ariaHidden":132},[158],[108,3056,3058,3061],{"className":3057},[162],[108,3059],{"className":3060,"style":557},[166],[108,3062,417],{"className":3063},[171,489]," lanzamientos es precisamente ",[108,3066,3068,3089],{"className":3067},[111],[108,3069,3071],{"className":3070},[115],[117,3072,3073],{"xmlns":119},[121,3074,3075,3087],{},[124,3076,3077,3079,3085],{},[130,3078,409],{"fence":132},[411,3080,3081,3083],{"linethickness":413},[415,3082,417],{},[415,3084,420],{},[130,3086,423],{"fence":132},[151,3088,426],{"encoding":153},[108,3090,3092],{"className":3091,"ariaHidden":132},[158],[108,3093,3095,3098],{"className":3094},[162],[108,3096],{"className":3097,"style":436},[166],[108,3099,3101,3107,3158],{"className":3100},[171],[108,3102,3104],{"className":3103,"style":445},[443,444],[108,3105,409],{"className":3106},[449,450],[108,3108,3110],{"className":3109},[411],[108,3111,3113,3150],{"className":3112},[457,458],[108,3114,3116,3147],{"className":3115},[462],[108,3117,3119,3133],{"className":3118,"style":467},[466],[108,3120,3121,3124],{"style":470},[108,3122],{"className":3123,"style":475},[474],[108,3125,3127],{"className":3126},[479,480,481,482],[108,3128,3130],{"className":3129},[171,482],[108,3131,420],{"className":3132,"style":490},[171,489,482],[108,3134,3135,3138],{"style":493},[108,3136],{"className":3137,"style":475},[474],[108,3139,3141],{"className":3140},[479,480,481,482],[108,3142,3144],{"className":3143},[171,482],[108,3145,417],{"className":3146},[171,489,482],[108,3148,510],{"className":3149},[509],[108,3151,3153],{"className":3152},[462],[108,3154,3156],{"className":3155,"style":517},[466],[108,3157],{},[108,3159,3161],{"className":3160,"style":445},[523,444],[108,3162,423],{"className":3163},[449,450]," — el coeficiente binomial que ocupa la posición ",[108,3166,3168,3181],{"className":3167},[111],[108,3169,3171],{"className":3170},[115],[117,3172,3173],{"xmlns":119},[121,3174,3175,3179],{},[124,3176,3177],{},[415,3178,420],{},[151,3180,420],{"encoding":153},[108,3182,3184],{"className":3183,"ariaHidden":132},[158],[108,3185,3187,3190],{"className":3186},[162],[108,3188],{"className":3189,"style":587},[166],[108,3191,420],{"className":3192,"style":490},[171,489]," en la fila ",[108,3195,3197,3210],{"className":3196},[111],[108,3198,3200],{"className":3199},[115],[117,3201,3202],{"xmlns":119},[121,3203,3204,3208],{},[124,3205,3206],{},[415,3207,417],{},[151,3209,417],{"encoding":153},[108,3211,3213],{"className":3212,"ariaHidden":132},[158],[108,3214,3216,3219],{"className":3215},[162],[108,3217],{"className":3218,"style":557},[166],[108,3220,417],{"className":3221},[171,489]," del triángulo.",[380,3224,3226],{"id":3225},"la-probabilidad-entra-en-escena","La probabilidad entra en escena",[15,3228,2265,3229,3232],{},[27,3230,3231],{},"probabilidad"," de un suceso mide qué tan posible es que ocurra. Se calcula como:",[15,3234,3236],{"className":3235},[394],[108,3237,3239,3264],{"className":3238},[111],[108,3240,3242],{"className":3241},[115],[117,3243,3244],{"xmlns":119},[121,3245,3246,3261],{},[124,3247,3248,3251,3253],{},[415,3249,3250],{},"P",[130,3252,680],{},[411,3254,3255,3258],{},[1886,3256,3257],{},"casos favorables",[1886,3259,3260],{},"casos posibles",[151,3262,3263],{"encoding":153},"P = \\frac{\\text{casos favorables}}{\\text{casos posibles}}",[108,3265,3267,3287],{"className":3266,"ariaHidden":132},[158],[108,3268,3270,3274,3278,3281,3284],{"className":3269},[162],[108,3271],{"className":3272,"style":3273},[166],"height:0.6833em;",[108,3275,3250],{"className":3276,"style":3277},[171,489],"margin-right:0.1389em;",[108,3279],{"className":3280,"style":766},[179],[108,3282,680],{"className":3283},[770],[108,3285],{"className":3286,"style":766},[179],[108,3288,3290,3294],{"className":3289},[162],[108,3291],{"className":3292,"style":3293},[166],"height:1.3612em;vertical-align:-0.4811em;",[108,3295,3297,3300,3366],{"className":3296},[171],[108,3298],{"className":3299},[443,977],[108,3301,3303],{"className":3302},[411],[108,3304,3306,3357],{"className":3305},[457,458],[108,3307,3309,3354],{"className":3308},[462],[108,3310,3312,3329,3337],{"className":3311,"style":990},[466],[108,3313,3314,3317],{"style":993},[108,3315],{"className":3316,"style":997},[474],[108,3318,3320],{"className":3319},[479,480,481,482],[108,3321,3323],{"className":3322},[171,482],[108,3324,3326],{"className":3325},[171,89,482],[108,3327,3260],{"className":3328},[171,482],[108,3330,3331,3334],{"style":1031},[108,3332],{"className":3333,"style":997},[474],[108,3335],{"className":3336,"style":1039},[1038],[108,3338,3339,3342],{"style":1042},[108,3340],{"className":3341,"style":997},[474],[108,3343,3345],{"className":3344},[479,480,481,482],[108,3346,3348],{"className":3347},[171,482],[108,3349,3351],{"className":3350},[171,89,482],[108,3352,3257],{"className":3353},[171,482],[108,3355,510],{"className":3356},[509],[108,3358,3360],{"className":3359},[462],[108,3361,3364],{"className":3362,"style":3363},[466],"height:0.4811em;",[108,3365],{},[108,3367],{"className":3368},[523,977],[15,3370,3371,3372,3400],{},"Si la moneda es justa, cada uno de los 8 resultados es igual de probable. Así que la probabilidad de obtener exactamente ",[108,3373,3375,3388],{"className":3374},[111],[108,3376,3378],{"className":3377},[115],[117,3379,3380],{"xmlns":119},[121,3381,3382,3386],{},[124,3383,3384],{},[415,3385,420],{},[151,3387,420],{"encoding":153},[108,3389,3391],{"className":3390,"ariaHidden":132},[158],[108,3392,3394,3397],{"className":3393},[162],[108,3395],{"className":3396,"style":587},[166],[108,3398,420],{"className":3399,"style":490},[171,489]," caras en 3 lanzamientos es:",[15,3402,3404],{"className":3403},[394],[108,3405,3407,3470],{"className":3406},[111],[108,3408,3410],{"className":3409},[115],[117,3411,3412],{"xmlns":119},[121,3413,3414,3467],{},[124,3415,3416,3418,3420,3422,3425,3427,3429,3449,3451],{},[415,3417,3250],{},[130,3419,409],{"stretchy":853},[415,3421,420],{},[1886,3423,3424],{}," caras",[130,3426,423],{"stretchy":853},[130,3428,680],{},[411,3430,3431,3443],{},[124,3432,3433,3435,3441],{},[130,3434,409],{"fence":132},[411,3436,3437,3439],{"linethickness":413},[127,3438,256],{},[415,3440,420],{},[130,3442,423],{"fence":132},[2506,3444,3445,3447],{},[127,3446,675],{},[127,3448,256],{},[130,3450,680],{},[411,3452,3453,3465],{},[124,3454,3455,3457,3463],{},[130,3456,409],{"fence":132},[411,3458,3459,3461],{"linethickness":413},[127,3460,256],{},[415,3462,420],{},[130,3464,423],{"fence":132},[127,3466,2413],{},[151,3468,3469],{"encoding":153},"P(k \\text{ caras}) = \\frac{\\binom{3}{k}}{2^3} = \\frac{\\binom{3}{k}}{8}",[108,3471,3473,3507,3698],{"className":3472,"ariaHidden":132},[158],[108,3474,3476,3480,3483,3486,3489,3495,3498,3501,3504],{"className":3475},[162],[108,3477],{"className":3478,"style":3479},[166],"height:1em;vertical-align:-0.25em;",[108,3481,3250],{"className":3482,"style":3277},[171,489],[108,3484,409],{"className":3485},[443],[108,3487,420],{"className":3488,"style":490},[171,489],[108,3490,3492],{"className":3491},[171,89],[108,3493,3424],{"className":3494},[171],[108,3496,423],{"className":3497},[523],[108,3499],{"className":3500,"style":766},[179],[108,3502,680],{"className":3503},[770],[108,3505],{"className":3506,"style":766},[179],[108,3508,3510,3514,3689,3692,3695],{"className":3509},[162],[108,3511],{"className":3512,"style":3513},[166],"height:1.855em;vertical-align:-0.345em;",[108,3515,3517,3520,3686],{"className":3516},[171],[108,3518],{"className":3519},[443,977],[108,3521,3523],{"className":3522},[411],[108,3524,3526,3678],{"className":3525},[457,458],[108,3527,3529,3675],{"className":3528},[462],[108,3530,3533,3577,3585],{"className":3531,"style":3532},[466],"height:1.51em;",[108,3534,3535,3538],{"style":993},[108,3536],{"className":3537,"style":997},[474],[108,3539,3541],{"className":3540},[479,480,481,482],[108,3542,3544],{"className":3543},[171,482],[108,3545,3547,3550],{"className":3546},[171,482],[108,3548,675],{"className":3549},[171,482],[108,3551,3553],{"className":3552},[2534],[108,3554,3556],{"className":3555},[457],[108,3557,3559],{"className":3558},[462],[108,3560,3563],{"className":3561,"style":3562},[466],"height:0.7463em;",[108,3564,3566,3570],{"style":3565},"top:-2.786em;margin-right:0.0714em;",[108,3567],{"className":3568,"style":3569},[474],"height:2.5em;",[108,3571,3574],{"className":3572},[479,3573,450,482],"reset-size3",[108,3575,256],{"className":3576},[171,482],[108,3578,3579,3582],{"style":1031},[108,3580],{"className":3581,"style":997},[474],[108,3583],{"className":3584,"style":1039},[1038],[108,3586,3588,3591],{"style":3587},"top:-3.7125em;",[108,3589],{"className":3590,"style":997},[474],[108,3592,3594],{"className":3593},[479,480,481,482],[108,3595,3597],{"className":3596},[171,482],[108,3598,3600,3611,3666],{"className":3599},[171,482],[108,3601,3605],{"className":3602,"style":3604},[443,479,3573,3603,482,444],"size6","top:0.075em;",[108,3606,3608],{"className":3607},[449,450,482],[108,3609,409],{"className":3610},[482],[108,3612,3614],{"className":3613},[411],[108,3615,3617,3657],{"className":3616},[457,458],[108,3618,3620,3654],{"className":3619},[462],[108,3621,3624,3639],{"className":3622,"style":3623},[466],"height:0.9313em;",[108,3625,3627,3630],{"style":3626},"top:-2.156em;",[108,3628],{"className":3629,"style":3569},[474],[108,3631,3633],{"className":3632},[479,3573,450,482],[108,3634,3636],{"className":3635},[171,482],[108,3637,420],{"className":3638,"style":490},[171,489,482],[108,3640,3642,3645],{"style":3641},"top:-2.971em;",[108,3643],{"className":3644,"style":3569},[474],[108,3646,3648],{"className":3647},[479,3573,450,482],[108,3649,3651],{"className":3650},[171,482],[108,3652,256],{"className":3653},[171,482],[108,3655,510],{"className":3656},[509],[108,3658,3660],{"className":3659},[462],[108,3661,3664],{"className":3662,"style":3663},[466],"height:0.344em;",[108,3665],{},[108,3667,3669],{"className":3668,"style":3604},[523,479,3573,3603,482,444],[108,3670,3672],{"className":3671},[449,450,482],[108,3673,423],{"className":3674},[482],[108,3676,510],{"className":3677},[509],[108,3679,3681],{"className":3680},[462],[108,3682,3684],{"className":3683,"style":517},[466],[108,3685],{},[108,3687],{"className":3688},[523,977],[108,3690],{"className":3691,"style":766},[179],[108,3693,680],{"className":3694},[770],[108,3696],{"className":3697,"style":766},[179],[108,3699,3701,3704],{"className":3700},[162],[108,3702],{"className":3703,"style":3513},[166],[108,3705,3707,3710,3838],{"className":3706},[171],[108,3708],{"className":3709},[443,977],[108,3711,3713],{"className":3712},[411],[108,3714,3716,3830],{"className":3715},[457,458],[108,3717,3719,3827],{"className":3718},[462],[108,3720,3722,3736,3744],{"className":3721,"style":3532},[466],[108,3723,3724,3727],{"style":993},[108,3725],{"className":3726,"style":997},[474],[108,3728,3730],{"className":3729},[479,480,481,482],[108,3731,3733],{"className":3732},[171,482],[108,3734,2413],{"className":3735},[171,482],[108,3737,3738,3741],{"style":1031},[108,3739],{"className":3740,"style":997},[474],[108,3742],{"className":3743,"style":1039},[1038],[108,3745,3746,3749],{"style":3587},[108,3747],{"className":3748,"style":997},[474],[108,3750,3752],{"className":3751},[479,480,481,482],[108,3753,3755],{"className":3754},[171,482],[108,3756,3758,3767,3818],{"className":3757},[171,482],[108,3759,3761],{"className":3760,"style":3604},[443,479,3573,3603,482,444],[108,3762,3764],{"className":3763},[449,450,482],[108,3765,409],{"className":3766},[482],[108,3768,3770],{"className":3769},[411],[108,3771,3773,3810],{"className":3772},[457,458],[108,3774,3776,3807],{"className":3775},[462],[108,3777,3779,3793],{"className":3778,"style":3623},[466],[108,3780,3781,3784],{"style":3626},[108,3782],{"className":3783,"style":3569},[474],[108,3785,3787],{"className":3786},[479,3573,450,482],[108,3788,3790],{"className":3789},[171,482],[108,3791,420],{"className":3792,"style":490},[171,489,482],[108,3794,3795,3798],{"style":3641},[108,3796],{"className":3797,"style":3569},[474],[108,3799,3801],{"className":3800},[479,3573,450,482],[108,3802,3804],{"className":3803},[171,482],[108,3805,256],{"className":3806},[171,482],[108,3808,510],{"className":3809},[509],[108,3811,3813],{"className":3812},[462],[108,3814,3816],{"className":3815,"style":3663},[466],[108,3817],{},[108,3819,3821],{"className":3820,"style":3604},[523,479,3573,3603,482,444],[108,3822,3824],{"className":3823},[449,450,482],[108,3825,423],{"className":3826},[482],[108,3828,510],{"className":3829},[509],[108,3831,3833],{"className":3832},[462],[108,3834,3836],{"className":3835,"style":517},[466],[108,3837],{},[108,3839],{"className":3840},[523,977],[15,3842,3843],{},"Calculemos cada caso:",[2618,3845,3846,3989],{},[2621,3847,3848],{},[2624,3849,3850,3881,3983,3986],{},[2627,3851,3852,3880],{"align":2629},[108,3853,3855,3868],{"className":3854},[111],[108,3856,3858],{"className":3857},[115],[117,3859,3860],{"xmlns":119},[121,3861,3862,3866],{},[124,3863,3864],{},[415,3865,420],{},[151,3867,420],{"encoding":153},[108,3869,3871],{"className":3870,"ariaHidden":132},[158],[108,3872,3874,3877],{"className":3873},[162],[108,3875],{"className":3876,"style":587},[166],[108,3878,420],{"className":3879,"style":490},[171,489]," caras",[2627,3882,3883],{"align":2629},[108,3884,3886,3908],{"className":3885},[111],[108,3887,3889],{"className":3888},[115],[117,3890,3891],{"xmlns":119},[121,3892,3893,3905],{},[124,3894,3895,3897,3903],{},[130,3896,409],{"fence":132},[411,3898,3899,3901],{"linethickness":413},[127,3900,256],{},[415,3902,420],{},[130,3904,423],{"fence":132},[151,3906,3907],{"encoding":153},"\\binom{3}{k}",[108,3909,3911],{"className":3910,"ariaHidden":132},[158],[108,3912,3914,3917],{"className":3913},[162],[108,3915],{"className":3916,"style":695},[166],[108,3918,3920,3926,3977],{"className":3919},[171],[108,3921,3923],{"className":3922,"style":445},[443,444],[108,3924,409],{"className":3925},[449,450],[108,3927,3929],{"className":3928},[411],[108,3930,3932,3969],{"className":3931},[457,458],[108,3933,3935,3966],{"className":3934},[462],[108,3936,3938,3952],{"className":3937,"style":717},[466],[108,3939,3940,3943],{"style":470},[108,3941],{"className":3942,"style":475},[474],[108,3944,3946],{"className":3945},[479,480,481,482],[108,3947,3949],{"className":3948},[171,482],[108,3950,420],{"className":3951,"style":490},[171,489,482],[108,3953,3954,3957],{"style":493},[108,3955],{"className":3956,"style":475},[474],[108,3958,3960],{"className":3959},[479,480,481,482],[108,3961,3963],{"className":3962},[171,482],[108,3964,256],{"className":3965},[171,482],[108,3967,510],{"className":3968},[509],[108,3970,3972],{"className":3971},[462],[108,3973,3975],{"className":3974,"style":517},[466],[108,3976],{},[108,3978,3980],{"className":3979,"style":445},[523,444],[108,3981,423],{"className":3982},[449,450],[2627,3984,3985],{"align":2629},"Probabilidad",[2627,3987,3988],{"align":2629},"Porcentaje",[2721,3990,3991,4133,4275,4413],{},[2624,3992,3993,3995,3997,4130],{},[2726,3994,1334],{"align":2629},[2726,3996,65],{"align":2629},[2726,3998,3999],{"align":2629},[108,4000,4002,4025],{"className":4001},[111],[108,4003,4005],{"className":4004},[115],[117,4006,4007],{"xmlns":119},[121,4008,4009,4022],{},[124,4010,4011,4017,4019],{},[411,4012,4013,4015],{},[127,4014,65],{},[127,4016,2413],{},[130,4018,680],{},[127,4020,4021],{},"0,125",[151,4023,4024],{"encoding":153},"\\frac{1}{8} = 0{,}125",[108,4026,4028,4111],{"className":4027,"ariaHidden":132},[158],[108,4029,4031,4034,4102,4105,4108],{"className":4030},[162],[108,4032],{"className":4033,"style":1736},[166],[108,4035,4037,4040,4099],{"className":4036},[171],[108,4038],{"className":4039},[443,977],[108,4041,4043],{"className":4042},[411],[108,4044,4046,4091],{"className":4045},[457,458],[108,4047,4049,4088],{"className":4048},[462],[108,4050,4052,4066,4074],{"className":4051,"style":1663},[466],[108,4053,4054,4057],{"style":993},[108,4055],{"className":4056,"style":997},[474],[108,4058,4060],{"className":4059},[479,480,481,482],[108,4061,4063],{"className":4062},[171,482],[108,4064,2413],{"className":4065},[171,482],[108,4067,4068,4071],{"style":1031},[108,4069],{"className":4070,"style":997},[474],[108,4072],{"className":4073,"style":1039},[1038],[108,4075,4076,4079],{"style":1042},[108,4077],{"className":4078,"style":997},[474],[108,4080,4082],{"className":4081},[479,480,481,482],[108,4083,4085],{"className":4084},[171,482],[108,4086,65],{"className":4087},[171,482],[108,4089,510],{"className":4090},[509],[108,4092,4094],{"className":4093},[462],[108,4095,4097],{"className":4096,"style":517},[466],[108,4098],{},[108,4100],{"className":4101},[523,977],[108,4103],{"className":4104,"style":766},[179],[108,4106,680],{"className":4107},[770],[108,4109],{"className":4110,"style":766},[179],[108,4112,4114,4117,4120,4126],{"className":4113},[162],[108,4115],{"className":4116,"style":167},[166],[108,4118,1334],{"className":4119},[171],[108,4121,4123],{"className":4122},[171],[108,4124,133],{"className":4125},[175],[108,4127,4129],{"className":4128},[171],"125",[2726,4131,4132],{"align":2629},"12,5 %",[2624,4134,4135,4137,4139,4272],{},[2726,4136,65],{"align":2629},[2726,4138,256],{"align":2629},[2726,4140,4141],{"align":2629},[108,4142,4144,4167],{"className":4143},[111],[108,4145,4147],{"className":4146},[115],[117,4148,4149],{"xmlns":119},[121,4150,4151,4164],{},[124,4152,4153,4159,4161],{},[411,4154,4155,4157],{},[127,4156,256],{},[127,4158,2413],{},[130,4160,680],{},[127,4162,4163],{},"0,375",[151,4165,4166],{"encoding":153},"\\frac{3}{8} = 0{,}375",[108,4168,4170,4253],{"className":4169,"ariaHidden":132},[158],[108,4171,4173,4176,4244,4247,4250],{"className":4172},[162],[108,4174],{"className":4175,"style":1736},[166],[108,4177,4179,4182,4241],{"className":4178},[171],[108,4180],{"className":4181},[443,977],[108,4183,4185],{"className":4184},[411],[108,4186,4188,4233],{"className":4187},[457,458],[108,4189,4191,4230],{"className":4190},[462],[108,4192,4194,4208,4216],{"className":4193,"style":1663},[466],[108,4195,4196,4199],{"style":993},[108,4197],{"className":4198,"style":997},[474],[108,4200,4202],{"className":4201},[479,480,481,482],[108,4203,4205],{"className":4204},[171,482],[108,4206,2413],{"className":4207},[171,482],[108,4209,4210,4213],{"style":1031},[108,4211],{"className":4212,"style":997},[474],[108,4214],{"className":4215,"style":1039},[1038],[108,4217,4218,4221],{"style":1042},[108,4219],{"className":4220,"style":997},[474],[108,4222,4224],{"className":4223},[479,480,481,482],[108,4225,4227],{"className":4226},[171,482],[108,4228,256],{"className":4229},[171,482],[108,4231,510],{"className":4232},[509],[108,4234,4236],{"className":4235},[462],[108,4237,4239],{"className":4238,"style":517},[466],[108,4240],{},[108,4242],{"className":4243},[523,977],[108,4245],{"className":4246,"style":766},[179],[108,4248,680],{"className":4249},[770],[108,4251],{"className":4252,"style":766},[179],[108,4254,4256,4259,4262,4268],{"className":4255},[162],[108,4257],{"className":4258,"style":167},[166],[108,4260,1334],{"className":4261},[171],[108,4263,4265],{"className":4264},[171],[108,4266,133],{"className":4267},[175],[108,4269,4271],{"className":4270},[171],"375",[2726,4273,4274],{"align":2629},"37,5 %",[2624,4276,4277,4279,4281,4411],{},[2726,4278,675],{"align":2629},[2726,4280,256],{"align":2629},[2726,4282,4283],{"align":2629},[108,4284,4286,4307],{"className":4285},[111],[108,4287,4289],{"className":4288},[115],[117,4290,4291],{"xmlns":119},[121,4292,4293,4305],{},[124,4294,4295,4301,4303],{},[411,4296,4297,4299],{},[127,4298,256],{},[127,4300,2413],{},[130,4302,680],{},[127,4304,4163],{},[151,4306,4166],{"encoding":153},[108,4308,4310,4393],{"className":4309,"ariaHidden":132},[158],[108,4311,4313,4316,4384,4387,4390],{"className":4312},[162],[108,4314],{"className":4315,"style":1736},[166],[108,4317,4319,4322,4381],{"className":4318},[171],[108,4320],{"className":4321},[443,977],[108,4323,4325],{"className":4324},[411],[108,4326,4328,4373],{"className":4327},[457,458],[108,4329,4331,4370],{"className":4330},[462],[108,4332,4334,4348,4356],{"className":4333,"style":1663},[466],[108,4335,4336,4339],{"style":993},[108,4337],{"className":4338,"style":997},[474],[108,4340,4342],{"className":4341},[479,480,481,482],[108,4343,4345],{"className":4344},[171,482],[108,4346,2413],{"className":4347},[171,482],[108,4349,4350,4353],{"style":1031},[108,4351],{"className":4352,"style":997},[474],[108,4354],{"className":4355,"style":1039},[1038],[108,4357,4358,4361],{"style":1042},[108,4359],{"className":4360,"style":997},[474],[108,4362,4364],{"className":4363},[479,480,481,482],[108,4365,4367],{"className":4366},[171,482],[108,4368,256],{"className":4369},[171,482],[108,4371,510],{"className":4372},[509],[108,4374,4376],{"className":4375},[462],[108,4377,4379],{"className":4378,"style":517},[466],[108,4380],{},[108,4382],{"className":4383},[523,977],[108,4385],{"className":4386,"style":766},[179],[108,4388,680],{"className":4389},[770],[108,4391],{"className":4392,"style":766},[179],[108,4394,4396,4399,4402,4408],{"className":4395},[162],[108,4397],{"className":4398,"style":167},[166],[108,4400,1334],{"className":4401},[171],[108,4403,4405],{"className":4404},[171],[108,4406,133],{"className":4407},[175],[108,4409,4271],{"className":4410},[171],[2726,4412,4274],{"align":2629},[2624,4414,4415,4417,4419,4549],{},[2726,4416,256],{"align":2629},[2726,4418,65],{"align":2629},[2726,4420,4421],{"align":2629},[108,4422,4424,4445],{"className":4423},[111],[108,4425,4427],{"className":4426},[115],[117,4428,4429],{"xmlns":119},[121,4430,4431,4443],{},[124,4432,4433,4439,4441],{},[411,4434,4435,4437],{},[127,4436,65],{},[127,4438,2413],{},[130,4440,680],{},[127,4442,4021],{},[151,4444,4024],{"encoding":153},[108,4446,4448,4531],{"className":4447,"ariaHidden":132},[158],[108,4449,4451,4454,4522,4525,4528],{"className":4450},[162],[108,4452],{"className":4453,"style":1736},[166],[108,4455,4457,4460,4519],{"className":4456},[171],[108,4458],{"className":4459},[443,977],[108,4461,4463],{"className":4462},[411],[108,4464,4466,4511],{"className":4465},[457,458],[108,4467,4469,4508],{"className":4468},[462],[108,4470,4472,4486,4494],{"className":4471,"style":1663},[466],[108,4473,4474,4477],{"style":993},[108,4475],{"className":4476,"style":997},[474],[108,4478,4480],{"className":4479},[479,480,481,482],[108,4481,4483],{"className":4482},[171,482],[108,4484,2413],{"className":4485},[171,482],[108,4487,4488,4491],{"style":1031},[108,4489],{"className":4490,"style":997},[474],[108,4492],{"className":4493,"style":1039},[1038],[108,4495,4496,4499],{"style":1042},[108,4497],{"className":4498,"style":997},[474],[108,4500,4502],{"className":4501},[479,480,481,482],[108,4503,4505],{"className":4504},[171,482],[108,4506,65],{"className":4507},[171,482],[108,4509,510],{"className":4510},[509],[108,4512,4514],{"className":4513},[462],[108,4515,4517],{"className":4516,"style":517},[466],[108,4518],{},[108,4520],{"className":4521},[523,977],[108,4523],{"className":4524,"style":766},[179],[108,4526,680],{"className":4527},[770],[108,4529],{"className":4530,"style":766},[179],[108,4532,4534,4537,4540,4546],{"className":4533},[162],[108,4535],{"className":4536,"style":167},[166],[108,4538,1334],{"className":4539},[171],[108,4541,4543],{"className":4542},[171],[108,4544,133],{"className":4545},[175],[108,4547,4129],{"className":4548},[171],[2726,4550,4132],{"align":2629},[15,4552,4553,4554,4707,4708,4736],{},"Si sumamos todas las probabilidades: ",[108,4555,4557,4587],{"className":4556},[111],[108,4558,4560],{"className":4559},[115],[117,4561,4562],{"xmlns":119},[121,4563,4564,4584],{},[124,4565,4566,4568,4570,4572,4574,4576,4578,4580,4582],{},[127,4567,4021],{},[130,4569,259],{},[127,4571,4163],{},[130,4573,259],{},[127,4575,4163],{},[130,4577,259],{},[127,4579,4021],{},[130,4581,680],{},[127,4583,65],{},[151,4585,4586],{"encoding":153},"0{,}125 + 0{,}375 + 0{,}375 + 0{,}125 = 1",[108,4588,4590,4617,4644,4671,4698],{"className":4589,"ariaHidden":132},[158],[108,4591,4593,4596,4599,4605,4608,4611,4614],{"className":4592},[162],[108,4594],{"className":4595,"style":167},[166],[108,4597,1334],{"className":4598},[171],[108,4600,4602],{"className":4601},[171],[108,4603,133],{"className":4604},[175],[108,4606,4129],{"className":4607},[171],[108,4609],{"className":4610,"style":281},[179],[108,4612,259],{"className":4613},[285],[108,4615],{"className":4616,"style":281},[179],[108,4618,4620,4623,4626,4632,4635,4638,4641],{"className":4619},[162],[108,4621],{"className":4622,"style":167},[166],[108,4624,1334],{"className":4625},[171],[108,4627,4629],{"className":4628},[171],[108,4630,133],{"className":4631},[175],[108,4633,4271],{"className":4634},[171],[108,4636],{"className":4637,"style":281},[179],[108,4639,259],{"className":4640},[285],[108,4642],{"className":4643,"style":281},[179],[108,4645,4647,4650,4653,4659,4662,4665,4668],{"className":4646},[162],[108,4648],{"className":4649,"style":167},[166],[108,4651,1334],{"className":4652},[171],[108,4654,4656],{"className":4655},[171],[108,4657,133],{"className":4658},[175],[108,4660,4271],{"className":4661},[171],[108,4663],{"className":4664,"style":281},[179],[108,4666,259],{"className":4667},[285],[108,4669],{"className":4670,"style":281},[179],[108,4672,4674,4677,4680,4686,4689,4692,4695],{"className":4673},[162],[108,4675],{"className":4676,"style":167},[166],[108,4678,1334],{"className":4679},[171],[108,4681,4683],{"className":4682},[171],[108,4684,133],{"className":4685},[175],[108,4687,4129],{"className":4688},[171],[108,4690],{"className":4691,"style":766},[179],[108,4693,680],{"className":4694},[770],[108,4696],{"className":4697,"style":766},[179],[108,4699,4701,4704],{"className":4700},[162],[108,4702],{"className":4703,"style":237},[166],[108,4705,65],{"className":4706},[171],". La suma siempre da ",[108,4709,4711,4724],{"className":4710},[111],[108,4712,4714],{"className":4713},[115],[117,4715,4716],{"xmlns":119},[121,4717,4718,4722],{},[124,4719,4720],{},[127,4721,65],{},[151,4723,65],{"encoding":153},[108,4725,4727],{"className":4726,"ariaHidden":132},[158],[108,4728,4730,4733],{"className":4729},[162],[108,4731],{"className":4732,"style":237},[166],[108,4734,65],{"className":4735},[171],", porque entre todos los casos se cubre el 100 % de las posibilidades.",[15,4738,4739],{},"La fórmula general que acabamos de descubrir funciona para cualquier número de lanzamientos:",[15,4741,4743],{"className":4742},[394],[108,4744,4746,4796],{"className":4745},[111],[108,4747,4749],{"className":4748},[115],[117,4750,4751],{"xmlns":119},[121,4752,4753,4793],{},[124,4754,4755,4757,4759,4761,4764,4766,4769,4771,4773],{},[415,4756,3250],{},[130,4758,409],{"stretchy":853},[415,4760,420],{},[1886,4762,4763],{}," caras en ",[415,4765,417],{},[1886,4767,4768],{}," lanzamientos",[130,4770,423],{"stretchy":853},[130,4772,680],{},[411,4774,4775,4787],{},[124,4776,4777,4779,4785],{},[130,4778,409],{"fence":132},[411,4780,4781,4783],{"linethickness":413},[415,4782,417],{},[415,4784,420],{},[130,4786,423],{"fence":132},[2506,4788,4789,4791],{},[127,4790,675],{},[415,4792,417],{},[151,4794,4795],{"encoding":153},"P(k \\text{ caras en } n \\text{ lanzamientos}) = \\frac{\\binom{n}{k}}{2^n}",[108,4797,4799,4841],{"className":4798,"ariaHidden":132},[158],[108,4800,4802,4805,4808,4811,4814,4820,4823,4829,4832,4835,4838],{"className":4801},[162],[108,4803],{"className":4804,"style":3479},[166],[108,4806,3250],{"className":4807,"style":3277},[171,489],[108,4809,409],{"className":4810},[443],[108,4812,420],{"className":4813,"style":490},[171,489],[108,4815,4817],{"className":4816},[171,89],[108,4818,4763],{"className":4819},[171],[108,4821,417],{"className":4822},[171,489],[108,4824,4826],{"className":4825},[171,89],[108,4827,4768],{"className":4828},[171],[108,4830,423],{"className":4831},[523],[108,4833],{"className":4834,"style":766},[179],[108,4836,680],{"className":4837},[770],[108,4839],{"className":4840,"style":766},[179],[108,4842,4844,4847],{"className":4843},[162],[108,4845],{"className":4846,"style":3513},[166],[108,4848,4850,4853,5009],{"className":4849},[171],[108,4851],{"className":4852},[443,977],[108,4854,4856],{"className":4855},[411],[108,4857,4859,5001],{"className":4858},[457,458],[108,4860,4862,4998],{"className":4861},[462],[108,4863,4865,4906,4914],{"className":4864,"style":3532},[466],[108,4866,4867,4870],{"style":993},[108,4868],{"className":4869,"style":997},[474],[108,4871,4873],{"className":4872},[479,480,481,482],[108,4874,4876],{"className":4875},[171,482],[108,4877,4879,4882],{"className":4878},[171,482],[108,4880,675],{"className":4881},[171,482],[108,4883,4885],{"className":4884},[2534],[108,4886,4888],{"className":4887},[457],[108,4889,4891],{"className":4890},[462],[108,4892,4895],{"className":4893,"style":4894},[466],"height:0.5935em;",[108,4896,4897,4900],{"style":3565},[108,4898],{"className":4899,"style":3569},[474],[108,4901,4903],{"className":4902},[479,3573,450,482],[108,4904,417],{"className":4905},[171,489,482],[108,4907,4908,4911],{"style":1031},[108,4909],{"className":4910,"style":997},[474],[108,4912],{"className":4913,"style":1039},[1038],[108,4915,4916,4919],{"style":3587},[108,4917],{"className":4918,"style":997},[474],[108,4920,4922],{"className":4921},[479,480,481,482],[108,4923,4925],{"className":4924},[171,482],[108,4926,4928,4937,4989],{"className":4927},[171,482],[108,4929,4931],{"className":4930,"style":3604},[443,479,3573,3603,482,444],[108,4932,4934],{"className":4933},[449,450,482],[108,4935,409],{"className":4936},[482],[108,4938,4940],{"className":4939},[411],[108,4941,4943,4981],{"className":4942},[457,458],[108,4944,4946,4978],{"className":4945},[462],[108,4947,4950,4964],{"className":4948,"style":4949},[466],"height:0.7785em;",[108,4951,4952,4955],{"style":3626},[108,4953],{"className":4954,"style":3569},[474],[108,4956,4958],{"className":4957},[479,3573,450,482],[108,4959,4961],{"className":4960},[171,482],[108,4962,420],{"className":4963,"style":490},[171,489,482],[108,4965,4966,4969],{"style":3641},[108,4967],{"className":4968,"style":3569},[474],[108,4970,4972],{"className":4971},[479,3573,450,482],[108,4973,4975],{"className":4974},[171,482],[108,4976,417],{"className":4977},[171,489,482],[108,4979,510],{"className":4980},[509],[108,4982,4984],{"className":4983},[462],[108,4985,4987],{"className":4986,"style":3663},[466],[108,4988],{},[108,4990,4992],{"className":4991,"style":3604},[523,479,3573,3603,482,444],[108,4993,4995],{"className":4994},[449,450,482],[108,4996,423],{"className":4997},[482],[108,4999,510],{"className":5000},[509],[108,5002,5004],{"className":5003},[462],[108,5005,5007],{"className":5006,"style":517},[466],[108,5008],{},[108,5010],{"className":5011},[523,977],[19,5013,5015],{"id":5014},"caso-práctico-5-lanzamientos","Caso práctico: 5 lanzamientos",[15,5017,5018],{},"Subamos la apuesta. Con 5 lanzamientos, la fila correspondiente del triángulo es:",[15,5020,5022],{"className":5021},[394],[108,5023,5025,5060],{"className":5024},[111],[108,5026,5028],{"className":5027},[115],[117,5029,5030],{"xmlns":119},[121,5031,5032,5057],{},[124,5033,5034,5036,5039,5041,5043,5045,5047,5049,5051,5053,5055],{},[127,5035,65],{},[179,5037],{"width":5038},"1em",[127,5040,2746],{},[179,5042],{"width":5038},[127,5044,2754],{},[179,5046],{"width":5038},[127,5048,2754],{},[179,5050],{"width":5038},[127,5052,2746],{},[179,5054],{"width":5038},[127,5056,65],{},[151,5058,5059],{"encoding":153},"1 \\quad 5 \\quad 10 \\quad 10 \\quad 5 \\quad 1",[108,5061,5063],{"className":5062,"ariaHidden":132},[158],[108,5064,5066,5069,5072,5076,5079,5082,5085,5088,5091,5094,5097,5100],{"className":5065},[162],[108,5067],{"className":5068,"style":237},[166],[108,5070,65],{"className":5071},[171],[108,5073],{"className":5074,"style":5075},[179],"margin-right:1em;",[108,5077,2746],{"className":5078},[171],[108,5080],{"className":5081,"style":5075},[179],[108,5083,2754],{"className":5084},[171],[108,5086],{"className":5087,"style":5075},[179],[108,5089,2754],{"className":5090},[171],[108,5092],{"className":5093,"style":5075},[179],[108,5095,2746],{"className":5096},[171],[108,5098],{"className":5099,"style":5075},[179],[108,5101,65],{"className":5102},[171],[15,5104,5105,5106,812],{},"y el total de resultados posibles es ",[108,5107,5109,5131],{"className":5108},[111],[108,5110,5112],{"className":5111},[115],[117,5113,5114],{"xmlns":119},[121,5115,5116,5128],{},[124,5117,5118,5124,5126],{},[2506,5119,5120,5122],{},[127,5121,675],{},[127,5123,2746],{},[130,5125,680],{},[127,5127,2749],{},[151,5129,5130],{"encoding":153},"2^5 = 32",[108,5132,5134,5178],{"className":5133,"ariaHidden":132},[158],[108,5135,5137,5140,5169,5172,5175],{"className":5136},[162],[108,5138],{"className":5139,"style":2812},[166],[108,5141,5143,5146],{"className":5142},[171],[108,5144,675],{"className":5145},[171],[108,5147,5149],{"className":5148},[2534],[108,5150,5152],{"className":5151},[457],[108,5153,5155],{"className":5154},[462],[108,5156,5158],{"className":5157,"style":2812},[466],[108,5159,5160,5163],{"style":2546},[108,5161],{"className":5162,"style":475},[474],[108,5164,5166],{"className":5165},[479,480,481,482],[108,5167,2746],{"className":5168},[171,482],[108,5170],{"className":5171,"style":766},[179],[108,5173,680],{"className":5174},[770],[108,5176],{"className":5177,"style":766},[179],[108,5179,5181,5184],{"className":5180},[162],[108,5182],{"className":5183,"style":237},[166],[108,5185,2749],{"className":5186},[171],[15,5188,5189,5190,5193,5194,5318],{},"Veamos algunos ejemplos concretos. ¿De cuántas formas podemos obtener exactamente ",[27,5191,5192],{},"2 caras","? El triángulo nos dice ",[108,5195,5197,5225],{"className":5196},[111],[108,5198,5200],{"className":5199},[115],[117,5201,5202],{"xmlns":119},[121,5203,5204,5222],{},[124,5205,5206,5218,5220],{},[124,5207,5208,5210,5216],{},[130,5209,409],{"fence":132},[411,5211,5212,5214],{"linethickness":413},[127,5213,2746],{},[127,5215,675],{},[130,5217,423],{"fence":132},[130,5219,680],{},[127,5221,2754],{},[151,5223,5224],{"encoding":153},"\\binom{5}{2} = 10",[108,5226,5228,5309],{"className":5227,"ariaHidden":132},[158],[108,5229,5231,5234,5300,5303,5306],{"className":5230},[162],[108,5232],{"className":5233,"style":695},[166],[108,5235,5237,5243,5294],{"className":5236},[171],[108,5238,5240],{"className":5239,"style":445},[443,444],[108,5241,409],{"className":5242},[449,450],[108,5244,5246],{"className":5245},[411],[108,5247,5249,5286],{"className":5248},[457,458],[108,5250,5252,5283],{"className":5251},[462],[108,5253,5255,5269],{"className":5254,"style":717},[466],[108,5256,5257,5260],{"style":470},[108,5258],{"className":5259,"style":475},[474],[108,5261,5263],{"className":5262},[479,480,481,482],[108,5264,5266],{"className":5265},[171,482],[108,5267,675],{"className":5268},[171,482],[108,5270,5271,5274],{"style":493},[108,5272],{"className":5273,"style":475},[474],[108,5275,5277],{"className":5276},[479,480,481,482],[108,5278,5280],{"className":5279},[171,482],[108,5281,2746],{"className":5282},[171,482],[108,5284,510],{"className":5285},[509],[108,5287,5289],{"className":5288},[462],[108,5290,5292],{"className":5291,"style":517},[466],[108,5293],{},[108,5295,5297],{"className":5296,"style":445},[523,444],[108,5298,423],{"className":5299},[449,450],[108,5301],{"className":5302,"style":766},[179],[108,5304,680],{"className":5305},[770],[108,5307],{"className":5308,"style":766},[179],[108,5310,5312,5315],{"className":5311},[162],[108,5313],{"className":5314,"style":237},[166],[108,5316,2754],{"className":5317},[171],". Estas son las 10 combinaciones:",[84,5320,5323],{"className":5321,"code":5322,"language":89,"meta":90},[87],"OOXXX OXOXX OXXOX OXXXO XOOXX\nXOXOX XOXXO XXOOX XXOXO XXXOO\n",[92,5324,5322],{"__ignoreMap":90},[15,5326,5327,5328,5331,5332,5456],{},"¿Y exactamente ",[27,5329,5330],{},"3 caras","? También ",[108,5333,5335,5363],{"className":5334},[111],[108,5336,5338],{"className":5337},[115],[117,5339,5340],{"xmlns":119},[121,5341,5342,5360],{},[124,5343,5344,5356,5358],{},[124,5345,5346,5348,5354],{},[130,5347,409],{"fence":132},[411,5349,5350,5352],{"linethickness":413},[127,5351,2746],{},[127,5353,256],{},[130,5355,423],{"fence":132},[130,5357,680],{},[127,5359,2754],{},[151,5361,5362],{"encoding":153},"\\binom{5}{3} = 10",[108,5364,5366,5447],{"className":5365,"ariaHidden":132},[158],[108,5367,5369,5372,5438,5441,5444],{"className":5368},[162],[108,5370],{"className":5371,"style":695},[166],[108,5373,5375,5381,5432],{"className":5374},[171],[108,5376,5378],{"className":5377,"style":445},[443,444],[108,5379,409],{"className":5380},[449,450],[108,5382,5384],{"className":5383},[411],[108,5385,5387,5424],{"className":5386},[457,458],[108,5388,5390,5421],{"className":5389},[462],[108,5391,5393,5407],{"className":5392,"style":717},[466],[108,5394,5395,5398],{"style":470},[108,5396],{"className":5397,"style":475},[474],[108,5399,5401],{"className":5400},[479,480,481,482],[108,5402,5404],{"className":5403},[171,482],[108,5405,256],{"className":5406},[171,482],[108,5408,5409,5412],{"style":493},[108,5410],{"className":5411,"style":475},[474],[108,5413,5415],{"className":5414},[479,480,481,482],[108,5416,5418],{"className":5417},[171,482],[108,5419,2746],{"className":5420},[171,482],[108,5422,510],{"className":5423},[509],[108,5425,5427],{"className":5426},[462],[108,5428,5430],{"className":5429,"style":517},[466],[108,5431],{},[108,5433,5435],{"className":5434,"style":445},[523,444],[108,5436,423],{"className":5437},[449,450],[108,5439],{"className":5440,"style":766},[179],[108,5442,680],{"className":5443},[770],[108,5445],{"className":5446,"style":766},[179],[108,5448,5450,5453],{"className":5449},[162],[108,5451],{"className":5452,"style":237},[166],[108,5454,2754],{"className":5455},[171],". El triángulo es simétrico: hay tantas formas de obtener 2 caras (y 3 cruces) como 3 caras (y 2 cruces).",[15,5458,5459],{},"La tabla completa de probabilidades:",[2618,5461,5462,5602],{},[2621,5463,5464],{},[2624,5465,5466,5496,5598,5600],{},[2627,5467,5468,3880],{"align":2629},[108,5469,5471,5484],{"className":5470},[111],[108,5472,5474],{"className":5473},[115],[117,5475,5476],{"xmlns":119},[121,5477,5478,5482],{},[124,5479,5480],{},[415,5481,420],{},[151,5483,420],{"encoding":153},[108,5485,5487],{"className":5486,"ariaHidden":132},[158],[108,5488,5490,5493],{"className":5489},[162],[108,5491],{"className":5492,"style":587},[166],[108,5494,420],{"className":5495,"style":490},[171,489],[2627,5497,5498],{"align":2629},[108,5499,5501,5523],{"className":5500},[111],[108,5502,5504],{"className":5503},[115],[117,5505,5506],{"xmlns":119},[121,5507,5508,5520],{},[124,5509,5510,5512,5518],{},[130,5511,409],{"fence":132},[411,5513,5514,5516],{"linethickness":413},[127,5515,2746],{},[415,5517,420],{},[130,5519,423],{"fence":132},[151,5521,5522],{"encoding":153},"\\binom{5}{k}",[108,5524,5526],{"className":5525,"ariaHidden":132},[158],[108,5527,5529,5532],{"className":5528},[162],[108,5530],{"className":5531,"style":695},[166],[108,5533,5535,5541,5592],{"className":5534},[171],[108,5536,5538],{"className":5537,"style":445},[443,444],[108,5539,409],{"className":5540},[449,450],[108,5542,5544],{"className":5543},[411],[108,5545,5547,5584],{"className":5546},[457,458],[108,5548,5550,5581],{"className":5549},[462],[108,5551,5553,5567],{"className":5552,"style":717},[466],[108,5554,5555,5558],{"style":470},[108,5556],{"className":5557,"style":475},[474],[108,5559,5561],{"className":5560},[479,480,481,482],[108,5562,5564],{"className":5563},[171,482],[108,5565,420],{"className":5566,"style":490},[171,489,482],[108,5568,5569,5572],{"style":493},[108,5570],{"className":5571,"style":475},[474],[108,5573,5575],{"className":5574},[479,480,481,482],[108,5576,5578],{"className":5577},[171,482],[108,5579,2746],{"className":5580},[171,482],[108,5582,510],{"className":5583},[509],[108,5585,5587],{"className":5586},[462],[108,5588,5590],{"className":5589,"style":517},[466],[108,5591],{},[108,5593,5595],{"className":5594,"style":445},[523,444],[108,5596,423],{"className":5597},[449,450],[2627,5599,3985],{"align":2629},[2627,5601,3988],{"align":2629},[2721,5603,5604,5713,5822,5931,6038,6145],{},[2624,5605,5606,5608,5610,5710],{},[2726,5607,1334],{"align":2629},[2726,5609,65],{"align":2629},[2726,5611,5612],{"align":2629},[108,5613,5615,5633],{"className":5614},[111],[108,5616,5618],{"className":5617},[115],[117,5619,5620],{"xmlns":119},[121,5621,5622,5630],{},[124,5623,5624],{},[411,5625,5626,5628],{},[127,5627,65],{},[127,5629,2749],{},[151,5631,5632],{"encoding":153},"\\frac{1}{32}",[108,5634,5636],{"className":5635,"ariaHidden":132},[158],[108,5637,5639,5642],{"className":5638},[162],[108,5640],{"className":5641,"style":1736},[166],[108,5643,5645,5648,5707],{"className":5644},[171],[108,5646],{"className":5647},[443,977],[108,5649,5651],{"className":5650},[411],[108,5652,5654,5699],{"className":5653},[457,458],[108,5655,5657,5696],{"className":5656},[462],[108,5658,5660,5674,5682],{"className":5659,"style":1663},[466],[108,5661,5662,5665],{"style":993},[108,5663],{"className":5664,"style":997},[474],[108,5666,5668],{"className":5667},[479,480,481,482],[108,5669,5671],{"className":5670},[171,482],[108,5672,2749],{"className":5673},[171,482],[108,5675,5676,5679],{"style":1031},[108,5677],{"className":5678,"style":997},[474],[108,5680],{"className":5681,"style":1039},[1038],[108,5683,5684,5687],{"style":1042},[108,5685],{"className":5686,"style":997},[474],[108,5688,5690],{"className":5689},[479,480,481,482],[108,5691,5693],{"className":5692},[171,482],[108,5694,65],{"className":5695},[171,482],[108,5697,510],{"className":5698},[509],[108,5700,5702],{"className":5701},[462],[108,5703,5705],{"className":5704,"style":517},[466],[108,5706],{},[108,5708],{"className":5709},[523,977],[2726,5711,5712],{"align":2629},"3,1 %",[2624,5714,5715,5717,5719,5819],{},[2726,5716,65],{"align":2629},[2726,5718,2746],{"align":2629},[2726,5720,5721],{"align":2629},[108,5722,5724,5742],{"className":5723},[111],[108,5725,5727],{"className":5726},[115],[117,5728,5729],{"xmlns":119},[121,5730,5731,5739],{},[124,5732,5733],{},[411,5734,5735,5737],{},[127,5736,2746],{},[127,5738,2749],{},[151,5740,5741],{"encoding":153},"\\frac{5}{32}",[108,5743,5745],{"className":5744,"ariaHidden":132},[158],[108,5746,5748,5751],{"className":5747},[162],[108,5749],{"className":5750,"style":1736},[166],[108,5752,5754,5757,5816],{"className":5753},[171],[108,5755],{"className":5756},[443,977],[108,5758,5760],{"className":5759},[411],[108,5761,5763,5808],{"className":5762},[457,458],[108,5764,5766,5805],{"className":5765},[462],[108,5767,5769,5783,5791],{"className":5768,"style":1663},[466],[108,5770,5771,5774],{"style":993},[108,5772],{"className":5773,"style":997},[474],[108,5775,5777],{"className":5776},[479,480,481,482],[108,5778,5780],{"className":5779},[171,482],[108,5781,2749],{"className":5782},[171,482],[108,5784,5785,5788],{"style":1031},[108,5786],{"className":5787,"style":997},[474],[108,5789],{"className":5790,"style":1039},[1038],[108,5792,5793,5796],{"style":1042},[108,5794],{"className":5795,"style":997},[474],[108,5797,5799],{"className":5798},[479,480,481,482],[108,5800,5802],{"className":5801},[171,482],[108,5803,2746],{"className":5804},[171,482],[108,5806,510],{"className":5807},[509],[108,5809,5811],{"className":5810},[462],[108,5812,5814],{"className":5813,"style":517},[466],[108,5815],{},[108,5817],{"className":5818},[523,977],[2726,5820,5821],{"align":2629},"15,6 %",[2624,5823,5824,5826,5828,5928],{},[2726,5825,675],{"align":2629},[2726,5827,2754],{"align":2629},[2726,5829,5830],{"align":2629},[108,5831,5833,5851],{"className":5832},[111],[108,5834,5836],{"className":5835},[115],[117,5837,5838],{"xmlns":119},[121,5839,5840,5848],{},[124,5841,5842],{},[411,5843,5844,5846],{},[127,5845,2754],{},[127,5847,2749],{},[151,5849,5850],{"encoding":153},"\\frac{10}{32}",[108,5852,5854],{"className":5853,"ariaHidden":132},[158],[108,5855,5857,5860],{"className":5856},[162],[108,5858],{"className":5859,"style":1736},[166],[108,5861,5863,5866,5925],{"className":5862},[171],[108,5864],{"className":5865},[443,977],[108,5867,5869],{"className":5868},[411],[108,5870,5872,5917],{"className":5871},[457,458],[108,5873,5875,5914],{"className":5874},[462],[108,5876,5878,5892,5900],{"className":5877,"style":1663},[466],[108,5879,5880,5883],{"style":993},[108,5881],{"className":5882,"style":997},[474],[108,5884,5886],{"className":5885},[479,480,481,482],[108,5887,5889],{"className":5888},[171,482],[108,5890,2749],{"className":5891},[171,482],[108,5893,5894,5897],{"style":1031},[108,5895],{"className":5896,"style":997},[474],[108,5898],{"className":5899,"style":1039},[1038],[108,5901,5902,5905],{"style":1042},[108,5903],{"className":5904,"style":997},[474],[108,5906,5908],{"className":5907},[479,480,481,482],[108,5909,5911],{"className":5910},[171,482],[108,5912,2754],{"className":5913},[171,482],[108,5915,510],{"className":5916},[509],[108,5918,5920],{"className":5919},[462],[108,5921,5923],{"className":5922,"style":517},[466],[108,5924],{},[108,5926],{"className":5927},[523,977],[2726,5929,5930],{"align":2629},"31,3 %",[2624,5932,5933,5935,5937,6036],{},[2726,5934,256],{"align":2629},[2726,5936,2754],{"align":2629},[2726,5938,5939],{"align":2629},[108,5940,5942,5959],{"className":5941},[111],[108,5943,5945],{"className":5944},[115],[117,5946,5947],{"xmlns":119},[121,5948,5949,5957],{},[124,5950,5951],{},[411,5952,5953,5955],{},[127,5954,2754],{},[127,5956,2749],{},[151,5958,5850],{"encoding":153},[108,5960,5962],{"className":5961,"ariaHidden":132},[158],[108,5963,5965,5968],{"className":5964},[162],[108,5966],{"className":5967,"style":1736},[166],[108,5969,5971,5974,6033],{"className":5970},[171],[108,5972],{"className":5973},[443,977],[108,5975,5977],{"className":5976},[411],[108,5978,5980,6025],{"className":5979},[457,458],[108,5981,5983,6022],{"className":5982},[462],[108,5984,5986,6000,6008],{"className":5985,"style":1663},[466],[108,5987,5988,5991],{"style":993},[108,5989],{"className":5990,"style":997},[474],[108,5992,5994],{"className":5993},[479,480,481,482],[108,5995,5997],{"className":5996},[171,482],[108,5998,2749],{"className":5999},[171,482],[108,6001,6002,6005],{"style":1031},[108,6003],{"className":6004,"style":997},[474],[108,6006],{"className":6007,"style":1039},[1038],[108,6009,6010,6013],{"style":1042},[108,6011],{"className":6012,"style":997},[474],[108,6014,6016],{"className":6015},[479,480,481,482],[108,6017,6019],{"className":6018},[171,482],[108,6020,2754],{"className":6021},[171,482],[108,6023,510],{"className":6024},[509],[108,6026,6028],{"className":6027},[462],[108,6029,6031],{"className":6030,"style":517},[466],[108,6032],{},[108,6034],{"className":6035},[523,977],[2726,6037,5930],{"align":2629},[2624,6039,6040,6042,6044,6143],{},[2726,6041,136],{"align":2629},[2726,6043,2746],{"align":2629},[2726,6045,6046],{"align":2629},[108,6047,6049,6066],{"className":6048},[111],[108,6050,6052],{"className":6051},[115],[117,6053,6054],{"xmlns":119},[121,6055,6056,6064],{},[124,6057,6058],{},[411,6059,6060,6062],{},[127,6061,2746],{},[127,6063,2749],{},[151,6065,5741],{"encoding":153},[108,6067,6069],{"className":6068,"ariaHidden":132},[158],[108,6070,6072,6075],{"className":6071},[162],[108,6073],{"className":6074,"style":1736},[166],[108,6076,6078,6081,6140],{"className":6077},[171],[108,6079],{"className":6080},[443,977],[108,6082,6084],{"className":6083},[411],[108,6085,6087,6132],{"className":6086},[457,458],[108,6088,6090,6129],{"className":6089},[462],[108,6091,6093,6107,6115],{"className":6092,"style":1663},[466],[108,6094,6095,6098],{"style":993},[108,6096],{"className":6097,"style":997},[474],[108,6099,6101],{"className":6100},[479,480,481,482],[108,6102,6104],{"className":6103},[171,482],[108,6105,2749],{"className":6106},[171,482],[108,6108,6109,6112],{"style":1031},[108,6110],{"className":6111,"style":997},[474],[108,6113],{"className":6114,"style":1039},[1038],[108,6116,6117,6120],{"style":1042},[108,6118],{"className":6119,"style":997},[474],[108,6121,6123],{"className":6122},[479,480,481,482],[108,6124,6126],{"className":6125},[171,482],[108,6127,2746],{"className":6128},[171,482],[108,6130,510],{"className":6131},[509],[108,6133,6135],{"className":6134},[462],[108,6136,6138],{"className":6137,"style":517},[466],[108,6139],{},[108,6141],{"className":6142},[523,977],[2726,6144,5821],{"align":2629},[2624,6146,6147,6149,6151,6250],{},[2726,6148,2746],{"align":2629},[2726,6150,65],{"align":2629},[2726,6152,6153],{"align":2629},[108,6154,6156,6173],{"className":6155},[111],[108,6157,6159],{"className":6158},[115],[117,6160,6161],{"xmlns":119},[121,6162,6163,6171],{},[124,6164,6165],{},[411,6166,6167,6169],{},[127,6168,65],{},[127,6170,2749],{},[151,6172,5632],{"encoding":153},[108,6174,6176],{"className":6175,"ariaHidden":132},[158],[108,6177,6179,6182],{"className":6178},[162],[108,6180],{"className":6181,"style":1736},[166],[108,6183,6185,6188,6247],{"className":6184},[171],[108,6186],{"className":6187},[443,977],[108,6189,6191],{"className":6190},[411],[108,6192,6194,6239],{"className":6193},[457,458],[108,6195,6197,6236],{"className":6196},[462],[108,6198,6200,6214,6222],{"className":6199,"style":1663},[466],[108,6201,6202,6205],{"style":993},[108,6203],{"className":6204,"style":997},[474],[108,6206,6208],{"className":6207},[479,480,481,482],[108,6209,6211],{"className":6210},[171,482],[108,6212,2749],{"className":6213},[171,482],[108,6215,6216,6219],{"style":1031},[108,6217],{"className":6218,"style":997},[474],[108,6220],{"className":6221,"style":1039},[1038],[108,6223,6224,6227],{"style":1042},[108,6225],{"className":6226,"style":997},[474],[108,6228,6230],{"className":6229},[479,480,481,482],[108,6231,6233],{"className":6232},[171,482],[108,6234,65],{"className":6235},[171,482],[108,6237,510],{"className":6238},[509],[108,6240,6242],{"className":6241},[462],[108,6243,6245],{"className":6244,"style":517},[466],[108,6246],{},[108,6248],{"className":6249},[523,977],[2726,6251,5712],{"align":2629},[15,6253,6254],{},"Si lo dibujamos como un histograma, la distribución tiene una forma familiar:",[84,6256,6259],{"className":6257,"code":6258,"language":89,"meta":90},[87],"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",[92,6260,6258],{"__ignoreMap":90},[15,6262,6263],{},"Observa tres cosas:",[57,6265,6266,6272,6278],{},[60,6267,6268,6271],{},[27,6269,6270],{},"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.",[60,6273,6274,6277],{},[27,6275,6276],{},"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.",[60,6279,6280,6283],{},[27,6281,6282],{},"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.",[15,6285,6286,6287,812],{},"Si te interesa explorar más la estadística detrás de los lanzamientos, lee ",[2765,6288,6291],{"href":6289,"rel":6290},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fblog\u002Fla-estadistica-inesperada-de-los-lanzamientos-de-moneda\u002F",[2769],"La estadística inesperada de los lanzamientos de moneda",[19,6293,6295],{"id":6294},"la-conexión-con-el-álgebra-el-binomio-de-newton","La conexión con el álgebra: el binomio de Newton",[15,6297,6298,6299,6302],{},"Hasta ahora hemos hablado de triángulos y monedas. Pero hay un tercer protagonista que conecta todo: el ",[27,6300,6301],{},"álgebra",". Concretamente, una expresión que quizá recuerdes del colegio:",[15,6304,6306],{"className":6305},[394],[108,6307,6309,6336],{"className":6308},[111],[108,6310,6312],{"className":6311},[115],[117,6313,6314],{"xmlns":119},[121,6315,6316,6333],{},[124,6317,6318,6320,6322,6324,6327],{},[130,6319,409],{"stretchy":853},[415,6321,2765],{},[130,6323,259],{},[415,6325,6326],{},"b",[2506,6328,6329,6331],{},[130,6330,423],{"stretchy":853},[415,6332,417],{},[151,6334,6335],{"encoding":153},"(a + b)^n",[108,6337,6339,6360],{"className":6338,"ariaHidden":132},[158],[108,6340,6342,6345,6348,6351,6354,6357],{"className":6341},[162],[108,6343],{"className":6344,"style":3479},[166],[108,6346,409],{"className":6347},[443],[108,6349,2765],{"className":6350},[171,489],[108,6352],{"className":6353,"style":281},[179],[108,6355,259],{"className":6356},[285],[108,6358],{"className":6359,"style":281},[179],[108,6361,6363,6366,6369],{"className":6362},[162],[108,6364],{"className":6365,"style":3479},[166],[108,6367,6326],{"className":6368},[171,489],[108,6370,6372,6375],{"className":6371},[523],[108,6373,423],{"className":6374},[523],[108,6376,6378],{"className":6377},[2534],[108,6379,6381],{"className":6380},[457],[108,6382,6384],{"className":6383},[462],[108,6385,6387],{"className":6386,"style":2524},[466],[108,6388,6389,6392],{"style":2546},[108,6390],{"className":6391,"style":475},[474],[108,6393,6395],{"className":6394},[479,480,481,482],[108,6396,417],{"className":6397},[171,489,482],[15,6399,6400],{},"Antes de desarrollarla, aclaremos un par de conceptos:",[6402,6403,6404,6526],"ul",{},[60,6405,6406,6407,6410,6411,812],{},"Un ",[27,6408,6409],{},"polinomio"," es una expresión matemática formada por sumas de términos con variables elevadas a distintas potencias. Por ejemplo: ",[108,6412,6414,6445],{"className":6413},[111],[108,6415,6417],{"className":6416},[115],[117,6418,6419],{"xmlns":119},[121,6420,6421,6442],{},[124,6422,6423,6425,6432,6434,6436,6438,6440],{},[127,6424,256],{},[2506,6426,6427,6430],{},[415,6428,6429],{},"x",[127,6431,675],{},[130,6433,259],{},[127,6435,675],{},[415,6437,6429],{},[130,6439,259],{},[127,6441,65],{},[151,6443,6444],{"encoding":153},"3x^2 + 2x + 1",[108,6446,6448,6496,6517],{"className":6447,"ariaHidden":132},[158],[108,6449,6451,6455,6458,6487,6490,6493],{"className":6450},[162],[108,6452],{"className":6453,"style":6454},[166],"height:0.8974em;vertical-align:-0.0833em;",[108,6456,256],{"className":6457},[171],[108,6459,6461,6464],{"className":6460},[171],[108,6462,6429],{"className":6463},[171,489],[108,6465,6467],{"className":6466},[2534],[108,6468,6470],{"className":6469},[457],[108,6471,6473],{"className":6472},[462],[108,6474,6476],{"className":6475,"style":2812},[466],[108,6477,6478,6481],{"style":2546},[108,6479],{"className":6480,"style":475},[474],[108,6482,6484],{"className":6483},[479,480,481,482],[108,6485,675],{"className":6486},[171,482],[108,6488],{"className":6489,"style":281},[179],[108,6491,259],{"className":6492},[285],[108,6494],{"className":6495,"style":281},[179],[108,6497,6499,6502,6505,6508,6511,6514],{"className":6498},[162],[108,6500],{"className":6501,"style":274},[166],[108,6503,675],{"className":6504},[171],[108,6506,6429],{"className":6507},[171,489],[108,6509],{"className":6510,"style":281},[179],[108,6512,259],{"className":6513},[285],[108,6515],{"className":6516,"style":281},[179],[108,6518,6520,6523],{"className":6519},[162],[108,6521],{"className":6522,"style":237},[166],[108,6524,65],{"className":6525},[171],[60,6527,6406,6528,6531,6532,6535,6536,812],{},[27,6529,6530],{},"binomio"," es un caso particular de polinomio: tiene exactamente ",[27,6533,6534],{},"dos"," términos. Por ejemplo: ",[108,6537,6539,6561],{"className":6538},[111],[108,6540,6542],{"className":6541},[115],[117,6543,6544],{"xmlns":119},[121,6545,6546,6558],{},[124,6547,6548,6550,6552,6554,6556],{},[130,6549,409],{"stretchy":853},[415,6551,2765],{},[130,6553,259],{},[415,6555,6326],{},[130,6557,423],{"stretchy":853},[151,6559,6560],{"encoding":153},"(a + b)",[108,6562,6564,6585],{"className":6563,"ariaHidden":132},[158],[108,6565,6567,6570,6573,6576,6579,6582],{"className":6566},[162],[108,6568],{"className":6569,"style":3479},[166],[108,6571,409],{"className":6572},[443],[108,6574,2765],{"className":6575},[171,489],[108,6577],{"className":6578,"style":281},[179],[108,6580,259],{"className":6581},[285],[108,6583],{"className":6584,"style":281},[179],[108,6586,6588,6591,6594],{"className":6587},[162],[108,6589],{"className":6590,"style":3479},[166],[108,6592,6326],{"className":6593},[171,489],[108,6595,423],{"className":6596},[523],[15,6598,6599,6600,6628],{},"Elevar un binomio a una potencia significa multiplicarlo por sí mismo ",[108,6601,6603,6616],{"className":6602},[111],[108,6604,6606],{"className":6605},[115],[117,6607,6608],{"xmlns":119},[121,6609,6610,6614],{},[124,6611,6612],{},[415,6613,417],{},[151,6615,417],{"encoding":153},[108,6617,6619],{"className":6618,"ariaHidden":132},[158],[108,6620,6622,6625],{"className":6621},[162],[108,6623],{"className":6624,"style":557},[166],[108,6626,417],{"className":6627},[171,489]," veces. Empecemos con los casos más sencillos:",[15,6630,6631],{},[27,6632,6633],{},"Potencia 2:",[15,6635,6637],{"className":6636},[394],[108,6638,6640,6690],{"className":6639},[111],[108,6641,6643],{"className":6642},[115],[117,6644,6645],{"xmlns":119},[121,6646,6647,6687],{},[124,6648,6649,6651,6653,6655,6657,6663,6665,6671,6673,6675,6677,6679,6681],{},[130,6650,409],{"stretchy":853},[415,6652,2765],{},[130,6654,259],{},[415,6656,6326],{},[2506,6658,6659,6661],{},[130,6660,423],{"stretchy":853},[127,6662,675],{},[130,6664,680],{},[2506,6666,6667,6669],{},[415,6668,2765],{},[127,6670,675],{},[130,6672,259],{},[127,6674,675],{},[415,6676,2765],{},[415,6678,6326],{},[130,6680,259],{},[2506,6682,6683,6685],{},[415,6684,6326],{},[127,6686,675],{},[151,6688,6689],{"encoding":153},"(a + b)^2 = a^2 + 2ab + b^2",[108,6691,6693,6714,6762,6806,6829],{"className":6692,"ariaHidden":132},[158],[108,6694,6696,6699,6702,6705,6708,6711],{"className":6695},[162],[108,6697],{"className":6698,"style":3479},[166],[108,6700,409],{"className":6701},[443],[108,6703,2765],{"className":6704},[171,489],[108,6706],{"className":6707,"style":281},[179],[108,6709,259],{"className":6710},[285],[108,6712],{"className":6713,"style":281},[179],[108,6715,6717,6721,6724,6753,6756,6759],{"className":6716},[162],[108,6718],{"className":6719,"style":6720},[166],"height:1.0641em;vertical-align:-0.25em;",[108,6722,6326],{"className":6723},[171,489],[108,6725,6727,6730],{"className":6726},[523],[108,6728,423],{"className":6729},[523],[108,6731,6733],{"className":6732},[2534],[108,6734,6736],{"className":6735},[457],[108,6737,6739],{"className":6738},[462],[108,6740,6742],{"className":6741,"style":2812},[466],[108,6743,6744,6747],{"style":2546},[108,6745],{"className":6746,"style":475},[474],[108,6748,6750],{"className":6749},[479,480,481,482],[108,6751,675],{"className":6752},[171,482],[108,6754],{"className":6755,"style":766},[179],[108,6757,680],{"className":6758},[770],[108,6760],{"className":6761,"style":766},[179],[108,6763,6765,6768,6797,6800,6803],{"className":6764},[162],[108,6766],{"className":6767,"style":6454},[166],[108,6769,6771,6774],{"className":6770},[171],[108,6772,2765],{"className":6773},[171,489],[108,6775,6777],{"className":6776},[2534],[108,6778,6780],{"className":6779},[457],[108,6781,6783],{"className":6782},[462],[108,6784,6786],{"className":6785,"style":2812},[466],[108,6787,6788,6791],{"style":2546},[108,6789],{"className":6790,"style":475},[474],[108,6792,6794],{"className":6793},[479,480,481,482],[108,6795,675],{"className":6796},[171,482],[108,6798],{"className":6799,"style":281},[179],[108,6801,259],{"className":6802},[285],[108,6804],{"className":6805,"style":281},[179],[108,6807,6809,6813,6816,6820,6823,6826],{"className":6808},[162],[108,6810],{"className":6811,"style":6812},[166],"height:0.7778em;vertical-align:-0.0833em;",[108,6814,675],{"className":6815},[171],[108,6817,6819],{"className":6818},[171,489],"ab",[108,6821],{"className":6822,"style":281},[179],[108,6824,259],{"className":6825},[285],[108,6827],{"className":6828,"style":281},[179],[108,6830,6832,6835],{"className":6831},[162],[108,6833],{"className":6834,"style":2812},[166],[108,6836,6838,6841],{"className":6837},[171],[108,6839,6326],{"className":6840},[171,489],[108,6842,6844],{"className":6843},[2534],[108,6845,6847],{"className":6846},[457],[108,6848,6850],{"className":6849},[462],[108,6851,6853],{"className":6852,"style":2812},[466],[108,6854,6855,6858],{"style":2546},[108,6856],{"className":6857,"style":475},[474],[108,6859,6861],{"className":6860},[479,480,481,482],[108,6862,675],{"className":6863},[171,482],[15,6865,6866,6867,6922],{},"Los coeficientes son ",[108,6868,6870,6892],{"className":6869},[111],[108,6871,6873],{"className":6872},[115],[117,6874,6875],{"xmlns":119},[121,6876,6877,6889],{},[124,6878,6879,6881,6883,6885,6887],{},[127,6880,65],{},[130,6882,133],{"separator":132},[127,6884,675],{},[130,6886,133],{"separator":132},[127,6888,65],{},[151,6890,6891],{"encoding":153},"1, 2, 1",[108,6893,6895],{"className":6894,"ariaHidden":132},[158],[108,6896,6898,6901,6904,6907,6910,6913,6916,6919],{"className":6897},[162],[108,6899],{"className":6900,"style":167},[166],[108,6902,65],{"className":6903},[171],[108,6905,133],{"className":6906},[175],[108,6908],{"className":6909,"style":180},[179],[108,6911,675],{"className":6912},[171],[108,6914,133],{"className":6915},[175],[108,6917],{"className":6918,"style":180},[179],[108,6920,65],{"className":6921},[171]," — la fila 2 del triángulo.",[15,6924,6925],{},[27,6926,6927],{},"Potencia 3:",[15,6929,6931],{"className":6930},[394],[108,6932,6934,7000],{"className":6933},[111],[108,6935,6937],{"className":6936},[115],[117,6938,6939],{"xmlns":119},[121,6940,6941,6997],{},[124,6942,6943,6945,6947,6949,6951,6957,6959,6965,6967,6969,6975,6977,6979,6981,6983,6989,6991],{},[130,6944,409],{"stretchy":853},[415,6946,2765],{},[130,6948,259],{},[415,6950,6326],{},[2506,6952,6953,6955],{},[130,6954,423],{"stretchy":853},[127,6956,256],{},[130,6958,680],{},[2506,6960,6961,6963],{},[415,6962,2765],{},[127,6964,256],{},[130,6966,259],{},[127,6968,256],{},[2506,6970,6971,6973],{},[415,6972,2765],{},[127,6974,675],{},[415,6976,6326],{},[130,6978,259],{},[127,6980,256],{},[415,6982,2765],{},[2506,6984,6985,6987],{},[415,6986,6326],{},[127,6988,675],{},[130,6990,259],{},[2506,6992,6993,6995],{},[415,6994,6326],{},[127,6996,256],{},[151,6998,6999],{"encoding":153},"(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3",[108,7001,7003,7024,7071,7115,7165,7215],{"className":7002,"ariaHidden":132},[158],[108,7004,7006,7009,7012,7015,7018,7021],{"className":7005},[162],[108,7007],{"className":7008,"style":3479},[166],[108,7010,409],{"className":7011},[443],[108,7013,2765],{"className":7014},[171,489],[108,7016],{"className":7017,"style":281},[179],[108,7019,259],{"className":7020},[285],[108,7022],{"className":7023,"style":281},[179],[108,7025,7027,7030,7033,7062,7065,7068],{"className":7026},[162],[108,7028],{"className":7029,"style":6720},[166],[108,7031,6326],{"className":7032},[171,489],[108,7034,7036,7039],{"className":7035},[523],[108,7037,423],{"className":7038},[523],[108,7040,7042],{"className":7041},[2534],[108,7043,7045],{"className":7044},[457],[108,7046,7048],{"className":7047},[462],[108,7049,7051],{"className":7050,"style":2812},[466],[108,7052,7053,7056],{"style":2546},[108,7054],{"className":7055,"style":475},[474],[108,7057,7059],{"className":7058},[479,480,481,482],[108,7060,256],{"className":7061},[171,482],[108,7063],{"className":7064,"style":766},[179],[108,7066,680],{"className":7067},[770],[108,7069],{"className":7070,"style":766},[179],[108,7072,7074,7077,7106,7109,7112],{"className":7073},[162],[108,7075],{"className":7076,"style":6454},[166],[108,7078,7080,7083],{"className":7079},[171],[108,7081,2765],{"className":7082},[171,489],[108,7084,7086],{"className":7085},[2534],[108,7087,7089],{"className":7088},[457],[108,7090,7092],{"className":7091},[462],[108,7093,7095],{"className":7094,"style":2812},[466],[108,7096,7097,7100],{"style":2546},[108,7098],{"className":7099,"style":475},[474],[108,7101,7103],{"className":7102},[479,480,481,482],[108,7104,256],{"className":7105},[171,482],[108,7107],{"className":7108,"style":281},[179],[108,7110,259],{"className":7111},[285],[108,7113],{"className":7114,"style":281},[179],[108,7116,7118,7121,7124,7153,7156,7159,7162],{"className":7117},[162],[108,7119],{"className":7120,"style":6454},[166],[108,7122,256],{"className":7123},[171],[108,7125,7127,7130],{"className":7126},[171],[108,7128,2765],{"className":7129},[171,489],[108,7131,7133],{"className":7132},[2534],[108,7134,7136],{"className":7135},[457],[108,7137,7139],{"className":7138},[462],[108,7140,7142],{"className":7141,"style":2812},[466],[108,7143,7144,7147],{"style":2546},[108,7145],{"className":7146,"style":475},[474],[108,7148,7150],{"className":7149},[479,480,481,482],[108,7151,675],{"className":7152},[171,482],[108,7154,6326],{"className":7155},[171,489],[108,7157],{"className":7158,"style":281},[179],[108,7160,259],{"className":7161},[285],[108,7163],{"className":7164,"style":281},[179],[108,7166,7168,7171,7174,7177,7206,7209,7212],{"className":7167},[162],[108,7169],{"className":7170,"style":6454},[166],[108,7172,256],{"className":7173},[171],[108,7175,2765],{"className":7176},[171,489],[108,7178,7180,7183],{"className":7179},[171],[108,7181,6326],{"className":7182},[171,489],[108,7184,7186],{"className":7185},[2534],[108,7187,7189],{"className":7188},[457],[108,7190,7192],{"className":7191},[462],[108,7193,7195],{"className":7194,"style":2812},[466],[108,7196,7197,7200],{"style":2546},[108,7198],{"className":7199,"style":475},[474],[108,7201,7203],{"className":7202},[479,480,481,482],[108,7204,675],{"className":7205},[171,482],[108,7207],{"className":7208,"style":281},[179],[108,7210,259],{"className":7211},[285],[108,7213],{"className":7214,"style":281},[179],[108,7216,7218,7221],{"className":7217},[162],[108,7219],{"className":7220,"style":2812},[166],[108,7222,7224,7227],{"className":7223},[171],[108,7225,6326],{"className":7226},[171,489],[108,7228,7230],{"className":7229},[2534],[108,7231,7233],{"className":7232},[457],[108,7234,7236],{"className":7235},[462],[108,7237,7239],{"className":7238,"style":2812},[466],[108,7240,7241,7244],{"style":2546},[108,7242],{"className":7243,"style":475},[474],[108,7245,7247],{"className":7246},[479,480,481,482],[108,7248,256],{"className":7249},[171,482],[15,7251,6866,7252,7319],{},[108,7253,7255,7280],{"className":7254},[111],[108,7256,7258],{"className":7257},[115],[117,7259,7260],{"xmlns":119},[121,7261,7262,7278],{},[124,7263,7264,7266,7268,7270,7272,7274,7276],{},[127,7265,65],{},[130,7267,133],{"separator":132},[127,7269,256],{},[130,7271,133],{"separator":132},[127,7273,256],{},[130,7275,133],{"separator":132},[127,7277,65],{},[151,7279,2959],{"encoding":153},[108,7281,7283],{"className":7282,"ariaHidden":132},[158],[108,7284,7286,7289,7292,7295,7298,7301,7304,7307,7310,7313,7316],{"className":7285},[162],[108,7287],{"className":7288,"style":167},[166],[108,7290,65],{"className":7291},[171],[108,7293,133],{"className":7294},[175],[108,7296],{"className":7297,"style":180},[179],[108,7299,256],{"className":7300},[171],[108,7302,133],{"className":7303},[175],[108,7305],{"className":7306,"style":180},[179],[108,7308,256],{"className":7309},[171],[108,7311,133],{"className":7312},[175],[108,7314],{"className":7315,"style":180},[179],[108,7317,65],{"className":7318},[171]," — la fila 3.",[15,7321,7322],{},[27,7323,7324],{},"Potencia 5:",[15,7326,7328],{"className":7327},[394],[108,7329,7331,7429],{"className":7330},[111],[108,7332,7334],{"className":7333},[115],[117,7335,7336],{"xmlns":119},[121,7337,7338,7426],{},[124,7339,7340,7342,7344,7346,7348,7354,7356,7362,7364,7366,7372,7374,7376,7378,7384,7390,7392,7394,7400,7406,7408,7410,7412,7418,7420],{},[130,7341,409],{"stretchy":853},[415,7343,2765],{},[130,7345,259],{},[415,7347,6326],{},[2506,7349,7350,7352],{},[130,7351,423],{"stretchy":853},[127,7353,2746],{},[130,7355,680],{},[2506,7357,7358,7360],{},[415,7359,2765],{},[127,7361,2746],{},[130,7363,259],{},[127,7365,2746],{},[2506,7367,7368,7370],{},[415,7369,2765],{},[127,7371,136],{},[415,7373,6326],{},[130,7375,259],{},[127,7377,2754],{},[2506,7379,7380,7382],{},[415,7381,2765],{},[127,7383,256],{},[2506,7385,7386,7388],{},[415,7387,6326],{},[127,7389,675],{},[130,7391,259],{},[127,7393,2754],{},[2506,7395,7396,7398],{},[415,7397,2765],{},[127,7399,675],{},[2506,7401,7402,7404],{},[415,7403,6326],{},[127,7405,256],{},[130,7407,259],{},[127,7409,2746],{},[415,7411,2765],{},[2506,7413,7414,7416],{},[415,7415,6326],{},[127,7417,136],{},[130,7419,259],{},[2506,7421,7422,7424],{},[415,7423,6326],{},[127,7425,2746],{},[151,7427,7428],{"encoding":153},"(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5",[108,7430,7432,7453,7500,7544,7594,7670,7746,7796],{"className":7431,"ariaHidden":132},[158],[108,7433,7435,7438,7441,7444,7447,7450],{"className":7434},[162],[108,7436],{"className":7437,"style":3479},[166],[108,7439,409],{"className":7440},[443],[108,7442,2765],{"className":7443},[171,489],[108,7445],{"className":7446,"style":281},[179],[108,7448,259],{"className":7449},[285],[108,7451],{"className":7452,"style":281},[179],[108,7454,7456,7459,7462,7491,7494,7497],{"className":7455},[162],[108,7457],{"className":7458,"style":6720},[166],[108,7460,6326],{"className":7461},[171,489],[108,7463,7465,7468],{"className":7464},[523],[108,7466,423],{"className":7467},[523],[108,7469,7471],{"className":7470},[2534],[108,7472,7474],{"className":7473},[457],[108,7475,7477],{"className":7476},[462],[108,7478,7480],{"className":7479,"style":2812},[466],[108,7481,7482,7485],{"style":2546},[108,7483],{"className":7484,"style":475},[474],[108,7486,7488],{"className":7487},[479,480,481,482],[108,7489,2746],{"className":7490},[171,482],[108,7492],{"className":7493,"style":766},[179],[108,7495,680],{"className":7496},[770],[108,7498],{"className":7499,"style":766},[179],[108,7501,7503,7506,7535,7538,7541],{"className":7502},[162],[108,7504],{"className":7505,"style":6454},[166],[108,7507,7509,7512],{"className":7508},[171],[108,7510,2765],{"className":7511},[171,489],[108,7513,7515],{"className":7514},[2534],[108,7516,7518],{"className":7517},[457],[108,7519,7521],{"className":7520},[462],[108,7522,7524],{"className":7523,"style":2812},[466],[108,7525,7526,7529],{"style":2546},[108,7527],{"className":7528,"style":475},[474],[108,7530,7532],{"className":7531},[479,480,481,482],[108,7533,2746],{"className":7534},[171,482],[108,7536],{"className":7537,"style":281},[179],[108,7539,259],{"className":7540},[285],[108,7542],{"className":7543,"style":281},[179],[108,7545,7547,7550,7553,7582,7585,7588,7591],{"className":7546},[162],[108,7548],{"className":7549,"style":6454},[166],[108,7551,2746],{"className":7552},[171],[108,7554,7556,7559],{"className":7555},[171],[108,7557,2765],{"className":7558},[171,489],[108,7560,7562],{"className":7561},[2534],[108,7563,7565],{"className":7564},[457],[108,7566,7568],{"className":7567},[462],[108,7569,7571],{"className":7570,"style":2812},[466],[108,7572,7573,7576],{"style":2546},[108,7574],{"className":7575,"style":475},[474],[108,7577,7579],{"className":7578},[479,480,481,482],[108,7580,136],{"className":7581},[171,482],[108,7583,6326],{"className":7584},[171,489],[108,7586],{"className":7587,"style":281},[179],[108,7589,259],{"className":7590},[285],[108,7592],{"className":7593,"style":281},[179],[108,7595,7597,7600,7603,7632,7661,7664,7667],{"className":7596},[162],[108,7598],{"className":7599,"style":6454},[166],[108,7601,2754],{"className":7602},[171],[108,7604,7606,7609],{"className":7605},[171],[108,7607,2765],{"className":7608},[171,489],[108,7610,7612],{"className":7611},[2534],[108,7613,7615],{"className":7614},[457],[108,7616,7618],{"className":7617},[462],[108,7619,7621],{"className":7620,"style":2812},[466],[108,7622,7623,7626],{"style":2546},[108,7624],{"className":7625,"style":475},[474],[108,7627,7629],{"className":7628},[479,480,481,482],[108,7630,256],{"className":7631},[171,482],[108,7633,7635,7638],{"className":7634},[171],[108,7636,6326],{"className":7637},[171,489],[108,7639,7641],{"className":7640},[2534],[108,7642,7644],{"className":7643},[457],[108,7645,7647],{"className":7646},[462],[108,7648,7650],{"className":7649,"style":2812},[466],[108,7651,7652,7655],{"style":2546},[108,7653],{"className":7654,"style":475},[474],[108,7656,7658],{"className":7657},[479,480,481,482],[108,7659,675],{"className":7660},[171,482],[108,7662],{"className":7663,"style":281},[179],[108,7665,259],{"className":7666},[285],[108,7668],{"className":7669,"style":281},[179],[108,7671,7673,7676,7679,7708,7737,7740,7743],{"className":7672},[162],[108,7674],{"className":7675,"style":6454},[166],[108,7677,2754],{"className":7678},[171],[108,7680,7682,7685],{"className":7681},[171],[108,7683,2765],{"className":7684},[171,489],[108,7686,7688],{"className":7687},[2534],[108,7689,7691],{"className":7690},[457],[108,7692,7694],{"className":7693},[462],[108,7695,7697],{"className":7696,"style":2812},[466],[108,7698,7699,7702],{"style":2546},[108,7700],{"className":7701,"style":475},[474],[108,7703,7705],{"className":7704},[479,480,481,482],[108,7706,675],{"className":7707},[171,482],[108,7709,7711,7714],{"className":7710},[171],[108,7712,6326],{"className":7713},[171,489],[108,7715,7717],{"className":7716},[2534],[108,7718,7720],{"className":7719},[457],[108,7721,7723],{"className":7722},[462],[108,7724,7726],{"className":7725,"style":2812},[466],[108,7727,7728,7731],{"style":2546},[108,7729],{"className":7730,"style":475},[474],[108,7732,7734],{"className":7733},[479,480,481,482],[108,7735,256],{"className":7736},[171,482],[108,7738],{"className":7739,"style":281},[179],[108,7741,259],{"className":7742},[285],[108,7744],{"className":7745,"style":281},[179],[108,7747,7749,7752,7755,7758,7787,7790,7793],{"className":7748},[162],[108,7750],{"className":7751,"style":6454},[166],[108,7753,2746],{"className":7754},[171],[108,7756,2765],{"className":7757},[171,489],[108,7759,7761,7764],{"className":7760},[171],[108,7762,6326],{"className":7763},[171,489],[108,7765,7767],{"className":7766},[2534],[108,7768,7770],{"className":7769},[457],[108,7771,7773],{"className":7772},[462],[108,7774,7776],{"className":7775,"style":2812},[466],[108,7777,7778,7781],{"style":2546},[108,7779],{"className":7780,"style":475},[474],[108,7782,7784],{"className":7783},[479,480,481,482],[108,7785,136],{"className":7786},[171,482],[108,7788],{"className":7789,"style":281},[179],[108,7791,259],{"className":7792},[285],[108,7794],{"className":7795,"style":281},[179],[108,7797,7799,7802],{"className":7798},[162],[108,7800],{"className":7801,"style":2812},[166],[108,7803,7805,7808],{"className":7804},[171],[108,7806,6326],{"className":7807},[171,489],[108,7809,7811],{"className":7810},[2534],[108,7812,7814],{"className":7813},[457],[108,7815,7817],{"className":7816},[462],[108,7818,7820],{"className":7819,"style":2812},[466],[108,7821,7822,7825],{"style":2546},[108,7823],{"className":7824,"style":475},[474],[108,7826,7828],{"className":7827},[479,480,481,482],[108,7829,2746],{"className":7830},[171,482],[15,7832,6866,7833,7927],{},[108,7834,7836,7870],{"className":7835},[111],[108,7837,7839],{"className":7838},[115],[117,7840,7841],{"xmlns":119},[121,7842,7843,7867],{},[124,7844,7845,7847,7849,7851,7853,7855,7857,7859,7861,7863,7865],{},[127,7846,65],{},[130,7848,133],{"separator":132},[127,7850,2746],{},[130,7852,133],{"separator":132},[127,7854,2754],{},[130,7856,133],{"separator":132},[127,7858,2754],{},[130,7860,133],{"separator":132},[127,7862,2746],{},[130,7864,133],{"separator":132},[127,7866,65],{},[151,7868,7869],{"encoding":153},"1, 5, 10, 10, 5, 1",[108,7871,7873],{"className":7872,"ariaHidden":132},[158],[108,7874,7876,7879,7882,7885,7888,7891,7894,7897,7900,7903,7906,7909,7912,7915,7918,7921,7924],{"className":7875},[162],[108,7877],{"className":7878,"style":167},[166],[108,7880,65],{"className":7881},[171],[108,7883,133],{"className":7884},[175],[108,7886],{"className":7887,"style":180},[179],[108,7889,2746],{"className":7890},[171],[108,7892,133],{"className":7893},[175],[108,7895],{"className":7896,"style":180},[179],[108,7898,2754],{"className":7899},[171],[108,7901,133],{"className":7902},[175],[108,7904],{"className":7905,"style":180},[179],[108,7907,2754],{"className":7908},[171],[108,7910,133],{"className":7911},[175],[108,7913],{"className":7914,"style":180},[179],[108,7916,2746],{"className":7917},[171],[108,7919,133],{"className":7920},[175],[108,7922],{"className":7923,"style":180},[179],[108,7925,65],{"className":7926},[171]," — la fila 5.",[15,7929,7930,7931,7934,7935,2489],{},"¿Ves el patrón? ",[27,7932,7933],{},"Las filas del triángulo de Tartaglia son exactamente los coeficientes del desarrollo del binomio."," Esta relación se formaliza con el ",[27,7936,7937],{},"teorema del binomio",[15,7939,7941],{"className":7940},[394],[108,7942,7944,8023],{"className":7943},[111],[108,7945,7947],{"className":7946},[115],[117,7948,7949],{"xmlns":119},[121,7950,7951,8020],{},[124,7952,7953,7955,7957,7959,7961,7967,7969,7985,7997,8000,8012,8014],{},[130,7954,409],{"stretchy":853},[415,7956,2765],{},[130,7958,259],{},[415,7960,6326],{},[2506,7962,7963,7965],{},[130,7964,423],{"stretchy":853},[415,7966,417],{},[130,7968,680],{},[7970,7971,7972,7975,7983],"msubsup",{},[130,7973,7974],{},"∑",[124,7976,7977,7979,7981],{},[415,7978,420],{},[130,7980,680],{},[127,7982,1334],{},[415,7984,417],{},[124,7986,7987,7989,7995],{},[130,7988,409],{"fence":132},[411,7990,7991,7993],{"linethickness":413},[415,7992,417],{},[415,7994,420],{},[130,7996,423],{"fence":132},[1886,7998,7999],{}," ",[2506,8001,8002,8004],{},[415,8003,2765],{},[124,8005,8006,8008,8010],{},[415,8007,417],{},[130,8009,870],{},[415,8011,420],{},[1886,8013,7999],{},[2506,8015,8016,8018],{},[415,8017,6326],{},[415,8019,420],{},[151,8021,8022],{"encoding":153},"(a+b)^n = \\sum_{k=0}^{n} \\binom{n}{k} \\, a^{n-k} \\, b^k",[108,8024,8026,8047,8094],{"className":8025,"ariaHidden":132},[158],[108,8027,8029,8032,8035,8038,8041,8044],{"className":8028},[162],[108,8030],{"className":8031,"style":3479},[166],[108,8033,409],{"className":8034},[443],[108,8036,2765],{"className":8037},[171,489],[108,8039],{"className":8040,"style":281},[179],[108,8042,259],{"className":8043},[285],[108,8045],{"className":8046,"style":281},[179],[108,8048,8050,8053,8056,8085,8088,8091],{"className":8049},[162],[108,8051],{"className":8052,"style":3479},[166],[108,8054,6326],{"className":8055},[171,489],[108,8057,8059,8062],{"className":8058},[523],[108,8060,423],{"className":8061},[523],[108,8063,8065],{"className":8064},[2534],[108,8066,8068],{"className":8067},[457],[108,8069,8071],{"className":8070},[462],[108,8072,8074],{"className":8073,"style":2524},[466],[108,8075,8076,8079],{"style":2546},[108,8077],{"className":8078,"style":475},[474],[108,8080,8082],{"className":8081},[479,480,481,482],[108,8083,417],{"className":8084},[171,489,482],[108,8086],{"className":8087,"style":766},[179],[108,8089,680],{"className":8090},[770],[108,8092],{"className":8093,"style":766},[179],[108,8095,8097,8100,8171,8174,8240,8243,8282,8285],{"className":8096},[162],[108,8098],{"className":8099,"style":436},[166],[108,8101,8104,8110],{"className":8102},[8103],"mop",[108,8105,7974],{"className":8106,"style":8109},[8103,8107,8108],"op-symbol","small-op","position:relative;top:0em;",[108,8111,8113],{"className":8112},[2534],[108,8114,8116,8162],{"className":8115},[457,458],[108,8117,8119,8159],{"className":8118},[462],[108,8120,8123,8144],{"className":8121,"style":8122},[466],"height:0.8043em;",[108,8124,8126,8129],{"style":8125},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[108,8127],{"className":8128,"style":475},[474],[108,8130,8132],{"className":8131},[479,480,481,482],[108,8133,8135,8138,8141],{"className":8134},[171,482],[108,8136,420],{"className":8137,"style":490},[171,489,482],[108,8139,680],{"className":8140},[770,482],[108,8142,1334],{"className":8143},[171,482],[108,8145,8147,8150],{"style":8146},"top:-3.2029em;margin-right:0.05em;",[108,8148],{"className":8149,"style":475},[474],[108,8151,8153],{"className":8152},[479,480,481,482],[108,8154,8156],{"className":8155},[171,482],[108,8157,417],{"className":8158},[171,489,482],[108,8160,510],{"className":8161},[509],[108,8163,8165],{"className":8164},[462],[108,8166,8169],{"className":8167,"style":8168},[466],"height:0.2997em;",[108,8170],{},[108,8172],{"className":8173,"style":180},[179],[108,8175,8177,8183,8234],{"className":8176},[171],[108,8178,8180],{"className":8179,"style":445},[443,444],[108,8181,409],{"className":8182},[449,450],[108,8184,8186],{"className":8185},[411],[108,8187,8189,8226],{"className":8188},[457,458],[108,8190,8192,8223],{"className":8191},[462],[108,8193,8195,8209],{"className":8194,"style":467},[466],[108,8196,8197,8200],{"style":470},[108,8198],{"className":8199,"style":475},[474],[108,8201,8203],{"className":8202},[479,480,481,482],[108,8204,8206],{"className":8205},[171,482],[108,8207,420],{"className":8208,"style":490},[171,489,482],[108,8210,8211,8214],{"style":493},[108,8212],{"className":8213,"style":475},[474],[108,8215,8217],{"className":8216},[479,480,481,482],[108,8218,8220],{"className":8219},[171,482],[108,8221,417],{"className":8222},[171,489,482],[108,8224,510],{"className":8225},[509],[108,8227,8229],{"className":8228},[462],[108,8230,8232],{"className":8231,"style":517},[466],[108,8233],{},[108,8235,8237],{"className":8236,"style":445},[523,444],[108,8238,423],{"className":8239},[449,450],[108,8241],{"className":8242,"style":180},[179],[108,8244,8246,8249],{"className":8245},[171],[108,8247,2765],{"className":8248},[171,489],[108,8250,8252],{"className":8251},[2534],[108,8253,8255],{"className":8254},[457],[108,8256,8258],{"className":8257},[462],[108,8259,8262],{"className":8260,"style":8261},[466],"height:0.8491em;",[108,8263,8264,8267],{"style":2546},[108,8265],{"className":8266,"style":475},[474],[108,8268,8270],{"className":8269},[479,480,481,482],[108,8271,8273,8276,8279],{"className":8272},[171,482],[108,8274,417],{"className":8275},[171,489,482],[108,8277,870],{"className":8278},[285,482],[108,8280,420],{"className":8281,"style":490},[171,489,482],[108,8283],{"className":8284,"style":180},[179],[108,8286,8288,8291],{"className":8287},[171],[108,8289,6326],{"className":8290},[171,489],[108,8292,8294],{"className":8293},[2534],[108,8295,8297],{"className":8296},[457],[108,8298,8300],{"className":8299},[462],[108,8301,8303],{"className":8302,"style":8261},[466],[108,8304,8305,8308],{"style":2546},[108,8306],{"className":8307,"style":475},[474],[108,8309,8311],{"className":8310},[479,480,481,482],[108,8312,420],{"className":8313,"style":490},[171,489,482],[15,8315,8316,8317,8346,8347,1143,8398,8449,8450,8549],{},"El símbolo ",[108,8318,8320,8334],{"className":8319},[111],[108,8321,8323],{"className":8322},[115],[117,8324,8325],{"xmlns":119},[121,8326,8327,8331],{},[124,8328,8329],{},[130,8330,7974],{},[151,8332,8333],{"encoding":153},"\\sum",[108,8335,8337],{"className":8336,"ariaHidden":132},[158],[108,8338,8340,8343],{"className":8339},[162],[108,8341],{"className":8342,"style":3479},[166],[108,8344,7974],{"className":8345,"style":8109},[8103,8107,8108]," (sigma) simplemente significa «suma todos los términos desde ",[108,8348,8350,8368],{"className":8349},[111],[108,8351,8353],{"className":8352},[115],[117,8354,8355],{"xmlns":119},[121,8356,8357,8365],{},[124,8358,8359,8361,8363],{},[415,8360,420],{},[130,8362,680],{},[127,8364,1334],{},[151,8366,8367],{"encoding":153},"k = 0",[108,8369,8371,8389],{"className":8370,"ariaHidden":132},[158],[108,8372,8374,8377,8380,8383,8386],{"className":8373},[162],[108,8375],{"className":8376,"style":587},[166],[108,8378,420],{"className":8379,"style":490},[171,489],[108,8381],{"className":8382,"style":766},[179],[108,8384,680],{"className":8385},[770],[108,8387],{"className":8388,"style":766},[179],[108,8390,8392,8395],{"className":8391},[162],[108,8393],{"className":8394,"style":237},[166],[108,8396,1334],{"className":8397},[171],[108,8399,8401,8419],{"className":8400},[111],[108,8402,8404],{"className":8403},[115],[117,8405,8406],{"xmlns":119},[121,8407,8408,8416],{},[124,8409,8410,8412,8414],{},[415,8411,420],{},[130,8413,680],{},[415,8415,417],{},[151,8417,8418],{"encoding":153},"k = n",[108,8420,8422,8440],{"className":8421,"ariaHidden":132},[158],[108,8423,8425,8428,8431,8434,8437],{"className":8424},[162],[108,8426],{"className":8427,"style":587},[166],[108,8429,420],{"className":8430,"style":490},[171,489],[108,8432],{"className":8433,"style":766},[179],[108,8435,680],{"className":8436},[770],[108,8438],{"className":8439,"style":766},[179],[108,8441,8443,8446],{"className":8442},[162],[108,8444],{"className":8445,"style":557},[166],[108,8447,417],{"className":8448},[171,489],"». Cada término tiene un coeficiente binomial ",[108,8451,8453,8474],{"className":8452},[111],[108,8454,8456],{"className":8455},[115],[117,8457,8458],{"xmlns":119},[121,8459,8460,8472],{},[124,8461,8462,8464,8470],{},[130,8463,409],{"fence":132},[411,8465,8466,8468],{"linethickness":413},[415,8467,417],{},[415,8469,420],{},[130,8471,423],{"fence":132},[151,8473,426],{"encoding":153},[108,8475,8477],{"className":8476,"ariaHidden":132},[158],[108,8478,8480,8483],{"className":8479},[162],[108,8481],{"className":8482,"style":436},[166],[108,8484,8486,8492,8543],{"className":8485},[171],[108,8487,8489],{"className":8488,"style":445},[443,444],[108,8490,409],{"className":8491},[449,450],[108,8493,8495],{"className":8494},[411],[108,8496,8498,8535],{"className":8497},[457,458],[108,8499,8501,8532],{"className":8500},[462],[108,8502,8504,8518],{"className":8503,"style":467},[466],[108,8505,8506,8509],{"style":470},[108,8507],{"className":8508,"style":475},[474],[108,8510,8512],{"className":8511},[479,480,481,482],[108,8513,8515],{"className":8514},[171,482],[108,8516,420],{"className":8517,"style":490},[171,489,482],[108,8519,8520,8523],{"style":493},[108,8521],{"className":8522,"style":475},[474],[108,8524,8526],{"className":8525},[479,480,481,482],[108,8527,8529],{"className":8528},[171,482],[108,8530,417],{"className":8531},[171,489,482],[108,8533,510],{"className":8534},[509],[108,8536,8538],{"className":8537},[462],[108,8539,8541],{"className":8540,"style":517},[466],[108,8542],{},[108,8544,8546],{"className":8545,"style":445},[523,444],[108,8547,423],{"className":8548},[449,450]," que viene directamente del triángulo.",[380,8551,8553],{"id":8552},"el-puente-entre-álgebra-y-probabilidad","El puente entre álgebra y probabilidad",[15,8555,8556,8557,8585,8586,8588,8589,8585,8617,2489],{},"Ahora viene lo fascinante. Sustituye ",[108,8558,8560,8573],{"className":8559},[111],[108,8561,8563],{"className":8562},[115],[117,8564,8565],{"xmlns":119},[121,8566,8567,8571],{},[124,8568,8569],{},[415,8570,2765],{},[151,8572,2765],{"encoding":153},[108,8574,8576],{"className":8575,"ariaHidden":132},[158],[108,8577,8579,8582],{"className":8578},[162],[108,8580],{"className":8581,"style":557},[166],[108,8583,2765],{"className":8584},[171,489]," por la probabilidad de obtener ",[27,8587,2278],{}," y ",[108,8590,8592,8605],{"className":8591},[111],[108,8593,8595],{"className":8594},[115],[117,8596,8597],{"xmlns":119},[121,8598,8599,8603],{},[124,8600,8601],{},[415,8602,6326],{},[151,8604,6326],{"encoding":153},[108,8606,8608],{"className":8607,"ariaHidden":132},[158],[108,8609,8611,8614],{"className":8610},[162],[108,8612],{"className":8613,"style":587},[166],[108,8615,6326],{"className":8616},[171,489],[27,8618,2282],{},[15,8620,8622],{"className":8621},[394],[108,8623,8625,8681],{"className":8624},[111],[108,8626,8628],{"className":8627},[115],[117,8629,8630],{"xmlns":119},[121,8631,8632,8678],{},[124,8633,8634,8636,8638,8640,8642,8644,8646,8648,8654,8656,8658,8660,8662,8664,8666,8668,8670,8672],{},[415,8635,2765],{},[130,8637,680],{},[415,8639,3250],{},[130,8641,409],{"stretchy":853},[1886,8643,2278],{},[130,8645,423],{"stretchy":853},[130,8647,680],{},[411,8649,8650,8652],{},[127,8651,65],{},[127,8653,675],{},[130,8655,133],{"separator":132},[179,8657],{"width":5038},[415,8659,6326],{},[130,8661,680],{},[415,8663,3250],{},[130,8665,409],{"stretchy":853},[1886,8667,2282],{},[130,8669,423],{"stretchy":853},[130,8671,680],{},[411,8673,8674,8676],{},[127,8675,65],{},[127,8677,675],{},[151,8679,8680],{"encoding":153},"a = P(\\text{cara}) = \\frac{1}{2}, \\quad b = P(\\text{cruz}) = \\frac{1}{2}",[108,8682,8684,8702,8732,8827,8857],{"className":8683,"ariaHidden":132},[158],[108,8685,8687,8690,8693,8696,8699],{"className":8686},[162],[108,8688],{"className":8689,"style":557},[166],[108,8691,2765],{"className":8692},[171,489],[108,8694],{"className":8695,"style":766},[179],[108,8697,680],{"className":8698},[770],[108,8700],{"className":8701,"style":766},[179],[108,8703,8705,8708,8711,8714,8720,8723,8726,8729],{"className":8704},[162],[108,8706],{"className":8707,"style":3479},[166],[108,8709,3250],{"className":8710,"style":3277},[171,489],[108,8712,409],{"className":8713},[443],[108,8715,8717],{"className":8716},[171,89],[108,8718,2278],{"className":8719},[171],[108,8721,423],{"className":8722},[523],[108,8724],{"className":8725,"style":766},[179],[108,8727,680],{"className":8728},[770],[108,8730],{"className":8731,"style":766},[179],[108,8733,8735,8738,8806,8809,8812,8815,8818,8821,8824],{"className":8734},[162],[108,8736],{"className":8737,"style":1736},[166],[108,8739,8741,8744,8803],{"className":8740},[171],[108,8742],{"className":8743},[443,977],[108,8745,8747],{"className":8746},[411],[108,8748,8750,8795],{"className":8749},[457,458],[108,8751,8753,8792],{"className":8752},[462],[108,8754,8756,8770,8778],{"className":8755,"style":1663},[466],[108,8757,8758,8761],{"style":993},[108,8759],{"className":8760,"style":997},[474],[108,8762,8764],{"className":8763},[479,480,481,482],[108,8765,8767],{"className":8766},[171,482],[108,8768,675],{"className":8769},[171,482],[108,8771,8772,8775],{"style":1031},[108,8773],{"className":8774,"style":997},[474],[108,8776],{"className":8777,"style":1039},[1038],[108,8779,8780,8783],{"style":1042},[108,8781],{"className":8782,"style":997},[474],[108,8784,8786],{"className":8785},[479,480,481,482],[108,8787,8789],{"className":8788},[171,482],[108,8790,65],{"className":8791},[171,482],[108,8793,510],{"className":8794},[509],[108,8796,8798],{"className":8797},[462],[108,8799,8801],{"className":8800,"style":517},[466],[108,8802],{},[108,8804],{"className":8805},[523,977],[108,8807,133],{"className":8808},[175],[108,8810],{"className":8811,"style":5075},[179],[108,8813],{"className":8814,"style":180},[179],[108,8816,6326],{"className":8817},[171,489],[108,8819],{"className":8820,"style":766},[179],[108,8822,680],{"className":8823},[770],[108,8825],{"className":8826,"style":766},[179],[108,8828,8830,8833,8836,8839,8845,8848,8851,8854],{"className":8829},[162],[108,8831],{"className":8832,"style":3479},[166],[108,8834,3250],{"className":8835,"style":3277},[171,489],[108,8837,409],{"className":8838},[443],[108,8840,8842],{"className":8841},[171,89],[108,8843,2282],{"className":8844},[171],[108,8846,423],{"className":8847},[523],[108,8849],{"className":8850,"style":766},[179],[108,8852,680],{"className":8853},[770],[108,8855],{"className":8856,"style":766},[179],[108,8858,8860,8863],{"className":8859},[162],[108,8861],{"className":8862,"style":1736},[166],[108,8864,8866,8869,8928],{"className":8865},[171],[108,8867],{"className":8868},[443,977],[108,8870,8872],{"className":8871},[411],[108,8873,8875,8920],{"className":8874},[457,458],[108,8876,8878,8917],{"className":8877},[462],[108,8879,8881,8895,8903],{"className":8880,"style":1663},[466],[108,8882,8883,8886],{"style":993},[108,8884],{"className":8885,"style":997},[474],[108,8887,8889],{"className":8888},[479,480,481,482],[108,8890,8892],{"className":8891},[171,482],[108,8893,675],{"className":8894},[171,482],[108,8896,8897,8900],{"style":1031},[108,8898],{"className":8899,"style":997},[474],[108,8901],{"className":8902,"style":1039},[1038],[108,8904,8905,8908],{"style":1042},[108,8906],{"className":8907,"style":997},[474],[108,8909,8911],{"className":8910},[479,480,481,482],[108,8912,8914],{"className":8913},[171,482],[108,8915,65],{"className":8916},[171,482],[108,8918,510],{"className":8919},[509],[108,8921,8923],{"className":8922},[462],[108,8924,8926],{"className":8925,"style":517},[466],[108,8927],{},[108,8929],{"className":8930},[523,977],[15,8932,8933],{},"El binomio se convierte en:",[15,8935,8937],{"className":8936},[394],[108,8938,8940,9078],{"className":8939},[111],[108,8941,8943],{"className":8942},[115],[117,8944,8945],{"xmlns":119},[121,8946,8947,9075],{},[124,8948,8949,8973,8975,8989,9001,9023,9039,9041,9055],{},[2506,8950,8951,8971],{},[124,8952,8953,8955,8961,8963,8969],{},[130,8954,409],{"fence":132},[411,8956,8957,8959],{},[127,8958,65],{},[127,8960,675],{},[130,8962,259],{},[411,8964,8965,8967],{},[127,8966,65],{},[127,8968,675],{},[130,8970,423],{"fence":132},[415,8972,417],{},[130,8974,680],{},[7970,8976,8977,8979,8987],{},[130,8978,7974],{},[124,8980,8981,8983,8985],{},[415,8982,420],{},[130,8984,680],{},[127,8986,1334],{},[415,8988,417],{},[124,8990,8991,8993,8999],{},[130,8992,409],{"fence":132},[411,8994,8995,8997],{"linethickness":413},[415,8996,417],{},[415,8998,420],{},[130,9000,423],{"fence":132},[2506,9002,9003,9015],{},[124,9004,9005,9007,9013],{},[130,9006,409],{"fence":132},[411,9008,9009,9011],{},[127,9010,65],{},[127,9012,675],{},[130,9014,423],{"fence":132},[124,9016,9017,9019,9021],{},[415,9018,417],{},[130,9020,870],{},[415,9022,420],{},[2506,9024,9025,9037],{},[124,9026,9027,9029,9035],{},[130,9028,409],{"fence":132},[411,9030,9031,9033],{},[127,9032,65],{},[127,9034,675],{},[130,9036,423],{"fence":132},[415,9038,420],{},[130,9040,680],{},[7970,9042,9043,9045,9053],{},[130,9044,7974],{},[124,9046,9047,9049,9051],{},[415,9048,420],{},[130,9050,680],{},[127,9052,1334],{},[415,9054,417],{},[411,9056,9057,9069],{},[124,9058,9059,9061,9067],{},[130,9060,409],{"fence":132},[411,9062,9063,9065],{"linethickness":413},[415,9064,417],{},[415,9066,420],{},[130,9068,423],{"fence":132},[2506,9070,9071,9073],{},[127,9072,675],{},[415,9074,417],{},[151,9076,9077],{"encoding":153},"\\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}",[108,9079,9081,9285,9667],{"className":9080,"ariaHidden":132},[158],[108,9082,9084,9088,9276,9279,9282],{"className":9083},[162],[108,9085],{"className":9086,"style":9087},[166],"height:1.2543em;vertical-align:-0.35em;",[108,9089,9091,9251],{"className":9090},[2180],[108,9092,9094,9100,9168,9171,9174,9177,9245],{"className":9093},[2180],[108,9095,9097],{"className":9096,"style":445},[443,444],[108,9098,409],{"className":9099},[449,450],[108,9101,9103,9106,9165],{"className":9102},[171],[108,9104],{"className":9105},[443,977],[108,9107,9109],{"className":9108},[411],[108,9110,9112,9157],{"className":9111},[457,458],[108,9113,9115,9154],{"className":9114},[462],[108,9116,9118,9132,9140],{"className":9117,"style":1663},[466],[108,9119,9120,9123],{"style":993},[108,9121],{"className":9122,"style":997},[474],[108,9124,9126],{"className":9125},[479,480,481,482],[108,9127,9129],{"className":9128},[171,482],[108,9130,675],{"className":9131},[171,482],[108,9133,9134,9137],{"style":1031},[108,9135],{"className":9136,"style":997},[474],[108,9138],{"className":9139,"style":1039},[1038],[108,9141,9142,9145],{"style":1042},[108,9143],{"className":9144,"style":997},[474],[108,9146,9148],{"className":9147},[479,480,481,482],[108,9149,9151],{"className":9150},[171,482],[108,9152,65],{"className":9153},[171,482],[108,9155,510],{"className":9156},[509],[108,9158,9160],{"className":9159},[462],[108,9161,9163],{"className":9162,"style":517},[466],[108,9164],{},[108,9166],{"className":9167},[523,977],[108,9169],{"className":9170,"style":281},[179],[108,9172,259],{"className":9173},[285],[108,9175],{"className":9176,"style":281},[179],[108,9178,9180,9183,9242],{"className":9179},[171],[108,9181],{"className":9182},[443,977],[108,9184,9186],{"className":9185},[411],[108,9187,9189,9234],{"className":9188},[457,458],[108,9190,9192,9231],{"className":9191},[462],[108,9193,9195,9209,9217],{"className":9194,"style":1663},[466],[108,9196,9197,9200],{"style":993},[108,9198],{"className":9199,"style":997},[474],[108,9201,9203],{"className":9202},[479,480,481,482],[108,9204,9206],{"className":9205},[171,482],[108,9207,675],{"className":9208},[171,482],[108,9210,9211,9214],{"style":1031},[108,9212],{"className":9213,"style":997},[474],[108,9215],{"className":9216,"style":1039},[1038],[108,9218,9219,9222],{"style":1042},[108,9220],{"className":9221,"style":997},[474],[108,9223,9225],{"className":9224},[479,480,481,482],[108,9226,9228],{"className":9227},[171,482],[108,9229,65],{"className":9230},[171,482],[108,9232,510],{"className":9233},[509],[108,9235,9237],{"className":9236},[462],[108,9238,9240],{"className":9239,"style":517},[466],[108,9241],{},[108,9243],{"className":9244},[523,977],[108,9246,9248],{"className":9247,"style":445},[523,444],[108,9249,423],{"className":9250},[449,450],[108,9252,9254],{"className":9253},[2534],[108,9255,9257],{"className":9256},[457],[108,9258,9260],{"className":9259},[462],[108,9261,9264],{"className":9262,"style":9263},[466],"height:0.9043em;",[108,9265,9267,9270],{"style":9266},"top:-3.3029em;margin-right:0.05em;",[108,9268],{"className":9269,"style":475},[474],[108,9271,9273],{"className":9272},[479,480,481,482],[108,9274,417],{"className":9275},[171,489,482],[108,9277],{"className":9278,"style":766},[179],[108,9280,680],{"className":9281},[770],[108,9283],{"className":9284,"style":766},[179],[108,9286,9288,9292,9355,9358,9424,9427,9546,9549,9658,9661,9664],{"className":9287},[162],[108,9289],{"className":9290,"style":9291},[166],"height:1.439em;vertical-align:-0.35em;",[108,9293,9295,9298],{"className":9294},[8103],[108,9296,7974],{"className":9297,"style":8109},[8103,8107,8108],[108,9299,9301],{"className":9300},[2534],[108,9302,9304,9347],{"className":9303},[457,458],[108,9305,9307,9344],{"className":9306},[462],[108,9308,9310,9330],{"className":9309,"style":8122},[466],[108,9311,9312,9315],{"style":8125},[108,9313],{"className":9314,"style":475},[474],[108,9316,9318],{"className":9317},[479,480,481,482],[108,9319,9321,9324,9327],{"className":9320},[171,482],[108,9322,420],{"className":9323,"style":490},[171,489,482],[108,9325,680],{"className":9326},[770,482],[108,9328,1334],{"className":9329},[171,482],[108,9331,9332,9335],{"style":8146},[108,9333],{"className":9334,"style":475},[474],[108,9336,9338],{"className":9337},[479,480,481,482],[108,9339,9341],{"className":9340},[171,482],[108,9342,417],{"className":9343},[171,489,482],[108,9345,510],{"className":9346},[509],[108,9348,9350],{"className":9349},[462],[108,9351,9353],{"className":9352,"style":8168},[466],[108,9354],{},[108,9356],{"className":9357,"style":180},[179],[108,9359,9361,9367,9418],{"className":9360},[171],[108,9362,9364],{"className":9363,"style":445},[443,444],[108,9365,409],{"className":9366},[449,450],[108,9368,9370],{"className":9369},[411],[108,9371,9373,9410],{"className":9372},[457,458],[108,9374,9376,9407],{"className":9375},[462],[108,9377,9379,9393],{"className":9378,"style":467},[466],[108,93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6],[108,9692,9693,9696],{"style":8125},[108,9694],{"className":9695,"style":475},[474],[108,9697,9699],{"className":9698},[479,480,481,482],[108,9700,9702,9705,9708],{"className":9701},[171,482],[108,9703,420],{"className":9704,"style":490},[171,489,482],[108,9706,680],{"className":9707},[770,482],[108,9709,1334],{"className":9710},[171,482],[108,9712,9713,9716],{"style":8146},[108,9714],{"className":9715,"style":475},[474],[108,9717,9719],{"className":9718},[479,480,481,482],[108,9720,9722],{"className":9721},[171,482],[108,9723,417],{"className":9724},[171,489,482],[108,9726,510],{"className":9727},[509],[108,9729,9731],{"className":9730},[462],[108,9732,9734],{"className":9733,"style":8168},[466],[108,9735],{},[108,9737],{"className":9738,"style":180},[179],[108,9740,9742,9745,9899],{"className":9741},[171],[108,9743],{"className":9744},[443,977],[108,9746,9748],{"className":9747},[411],[108,9749,9751,9891],{"className":9750},[457,458],[108,9752,9754,9888],{"className":9753},[462],[108,9755,9757,9797,9805],{"className":9756,"style":3532},[466],[108,9758,9759,9762],{"style":993},[108,9760],{"className":9761,"style":997},[474],[108,9763,9765],{"className":9764},[479,480,481,482],[108,9766,9768],{"className":9767},[171,482],[108,9769,9771,9774],{"className":9770},[171,482],[108,9772,675],{"className":9773},[171,482],[108,9775,9777],{"className":9776},[2534],[108,9778,9780],{"className":9779},[457],[108,9781,9783],{"className":9782},[462],[108,9784,9786],{"className":9785,"style":4894},[466],[108,9787,9788,9791],{"style":3565},[108,9789],{"className":9790,"style":3569},[474],[108,9792,9794],{"className":9793},[479,3573,450,482],[108,9795,417],{"className":9796},[171,489,482],[108,9798,9799,9802],{"style":1031},[108,9800],{"className":9801,"style":997},[474],[108,9803],{"className":9804,"style":1039},[1038],[108,9806,9807,9810],{"style":3587},[108,9808],{"className":9809,"style":997},[474],[108,9811,9813],{"className":9812},[479,480,481,482],[108,9814,9816],{"className":9815},[171,482],[108,9817,9819,9828,9879],{"className":9818},[171,482],[108,9820,9822],{"className":9821,"style":3604},[443,479,3573,3603,482,444],[108,9823,9825],{"className":9824},[449,450,482],[108,9826,409],{"className":9827},[482],[108,9829,9831],{"className":9830},[411],[108,9832,9834,9871],{"className":9833},[457,458],[108,9835,9837,9868],{"className":9836},[462],[108,9838,9840,9854],{"className":9839,"style":4949},[466],[108,9841,9842,9845],{"style":3626},[108,9843],{"className":9844,"style":3569},[474],[108,9846,9848],{"className":9847},[479,3573,450,482],[108,9849,9851],{"className":9850},[171,482],[108,9852,420],{"className":9853,"style":490},[171,489,482],[108,9855,9856,9859],{"style":3641},[108,9857],{"className":9858,"style":3569},[474],[108,9860,9862],{"className":9861},[479,3573,450,482],[108,9863,9865],{"className":9864},[171,482],[108,9866,417],{"className":9867},[171,489,482],[108,9869,510],{"className":9870},[509],[108,9872,9874],{"className":9873},[462],[108,9875,9877],{"className":9876,"style":3663},[466],[108,9878],{},[108,9880,9882],{"className":9881,"style":3604},[523,479,3573,3603,482,444],[108,9883,9885],{"className":9884},[449,450,482],[108,9886,423],{"className":9887},[482],[108,9889,510],{"className":9890},[509],[108,9892,9894],{"className":9893},[462],[108,9895,9897],{"className":9896,"style":517},[466],[108,9898],{},[108,9900],{"className":9901},[523,977],[15,9903,9904,9905,9986,9987,10194,10195,10223,10224,10276],{},"El lado izquierdo es simplemente ",[108,9906,9908,9930],{"className":9907},[111],[108,9909,9911],{"className":9910},[115],[117,9912,9913],{"xmlns":119},[121,9914,9915,9927],{},[124,9916,9917,9923,9925],{},[2506,9918,9919,9921],{},[127,9920,65],{},[415,9922,417],{},[130,9924,680],{},[127,9926,65],{},[151,9928,9929],{"encoding":153},"1^n = 1",[108,9931,9933,9977],{"className":9932,"ariaHidden":132},[158],[108,9934,9936,9939,9968,9971,9974],{"className":9935},[162],[108,9937],{"className":9938,"style":2524},[166],[108,9940,9942,9945],{"className":9941},[171],[108,9943,65],{"className":9944},[171],[108,9946,9948],{"className":9947},[2534],[108,9949,9951],{"className":9950},[457],[108,9952,9954],{"className":9953},[462],[108,9955,9957],{"className":9956,"style":2524},[466],[108,9958,9959,9962],{"style":2546},[108,9960],{"className":9961,"style":475},[474],[108,9963,9965],{"className":9964},[479,480,481,482],[108,9966,417],{"className":9967},[171,489,482],[108,9969],{"className":9970,"style":766},[179],[108,9972,680],{"className":9973},[770],[108,9975],{"className":9976,"style":766},[179],[108,9978,9980,9983],{"className":9979},[162],[108,9981],{"className":9982,"style":237},[166],[108,9984,65],{"className":9985},[171],": la certeza total. El lado derecho es la suma de todas las probabilidades individuales. Y cada término ",[108,9988,9990,10022],{"className":9989},[111],[108,9991,9993],{"className":9992},[115],[117,9994,9995],{"xmlns":119},[121,9996,9997,10019],{},[124,9998,9999],{},[411,10000,10001,10013],{},[124,10002,10003,10005,10011],{},[130,10004,409],{"fence":132},[411,10006,10007,10009],{"linethickness":413},[415,10008,417],{},[415,10010,420],{},[130,10012,423],{"fence":132},[2506,10014,10015,10017],{},[127,10016,675],{},[415,10018,417],{},[151,10020,10021],{"encoding":153},"\\frac{\\binom{n}{k}}{2^n}",[108,10023,10025],{"className":10024,"ariaHidden":132},[158],[108,10026,10028,10031],{"className":10027},[162],[108,10029],{"className":10030,"style":3513},[166],[108,10032,10034,10037,10191],{"className":10033},[171],[108,10035],{"className":10036},[443,977],[108,10038,10040],{"className":10039},[411],[108,10041,10043,10183],{"className":10042},[457,458],[108,10044,10046,10180],{"className":10045},[462],[108,10047,10049,10089,10097],{"className":10048,"style":3532},[466],[108,10050,10051,10054],{"style":993},[108,10052],{"className":10053,"style":997},[474],[108,10055,10057],{"className":10056},[479,480,481,482],[108,10058,10060],{"className":10059},[171,482],[108,10061,10063,10066],{"className":10062},[171,482],[108,10064,675],{"className":10065},[171,482],[108,10067,10069],{"className":10068},[2534],[108,10070,10072],{"className":10071},[457],[108,10073,10075],{"className":10074},[462],[108,10076,10078],{"className":10077,"style":4894},[466],[108,10079,10080,10083],{"style":3565},[108,10081],{"className":10082,"style":3569},[474],[108,10084,10086],{"className":10085},[479,3573,450,482],[108,10087,417],{"className":10088},[171,489,482],[108,10090,10091,10094],{"style":1031},[108,10092],{"className":10093,"style":997},[474],[108,10095],{"className":10096,"style":1039},[1038],[108,10098,10099,10102],{"style":3587},[108,10100],{"className":10101,"style":997},[474],[108,10103,10105],{"className":10104},[479,480,481,482],[108,10106,10108],{"className":10107},[171,482],[108,10109,10111,10120,10171],{"className":10110},[171,482],[108,10112,10114],{"className":10113,"style":3604},[443,479,3573,3603,482,444],[108,10115,10117],{"className":10116},[449,450,482],[108,10118,409],{"className":10119},[482],[108,10121,10123],{"className":10122},[411],[108,10124,10126,10163],{"className":10125},[457,458],[108,10127,10129,10160],{"className":10128},[462],[108,10130,10132,10146],{"className":10131,"style":4949},[466],[108,10133,10134,10137],{"style":3626},[108,10135],{"className":10136,"style":3569},[474],[108,10138,10140],{"className":10139},[479,3573,450,482],[108,10141,10143],{"className":10142},[171,482],[108,10144,420],{"className":10145,"style":490},[171,489,482],[108,10147,10148,10151],{"style":3641},[108,10149],{"className":10150,"style":3569},[474],[108,10152,10154],{"className":10153},[479,3573,450,482],[108,10155,10157],{"className":10156},[171,482],[108,10158,417],{"className":10159},[171,489,482],[108,10161,510],{"className":10162},[509],[108,10164,10166],{"className":10165},[462],[108,10167,10169],{"className":10168,"style":3663},[466],[108,10170],{},[108,10172,10174],{"className":10173,"style":3604},[523,479,3573,3603,482,444],[108,10175,10177],{"className":10176},[449,450,482],[108,10178,423],{"className":10179},[482],[108,10181,510],{"className":10182},[509],[108,10184,10186],{"className":10185},[462],[108,10187,10189],{"className":10188,"style":517},[466],[108,10190],{},[108,10192],{"className":10193},[523,977]," es exactamente la probabilidad de obtener ",[108,10196,10198,10211],{"className":10197},[111],[108,10199,10201],{"className":10200},[115],[117,10202,10203],{"xmlns":119},[121,10204,10205,10209],{},[124,10206,10207],{},[415,10208,420],{},[151,10210,420],{"encoding":153},[108,10212,10214],{"className":10213,"ariaHidden":132},[158],[108,10215,10217,10220],{"className":10216},[162],[108,10218],{"className":10219,"style":587},[166],[108,10221,420],{"className":10222,"style":490},[171,489]," cruces (o equivalentemente, ",[108,10225,10227,10245],{"className":10226},[111],[108,10228,10230],{"className":10229},[115],[117,10231,10232],{"xmlns":119},[121,10233,10234,10242],{},[124,10235,10236,10238,10240],{},[415,10237,417],{},[130,10239,870],{},[415,10241,420],{},[151,10243,10244],{"encoding":153},"n - k",[108,10246,10248,10267],{"className":10247,"ariaHidden":132},[158],[108,10249,10251,10255,10258,10261,10264],{"className":10250},[162],[108,10252],{"className":10253,"style":10254},[166],"height:0.6667em;vertical-align:-0.0833em;",[108,10256,417],{"className":10257},[171,489],[108,10259],{"className":10260,"style":281},[179],[108,10262,870],{"className":10263},[285],[108,10265],{"className":10266,"style":281},[179],[108,10268,10270,10273],{"className":10269},[162],[108,10271],{"className":10272,"style":587},[166],[108,10274,420],{"className":10275,"style":490},[171,489]," caras).",[15,10278,10279],{},"Comprobémoslo con 5 lanzamientos:",[15,10281,10283],{"className":10282},[394],[108,10284,10286,10338],{"className":10285},[111],[108,10287,10289],{"className":10288},[115],[117,10290,10291],{"xmlns":119},[121,10292,10293,10335],{},[124,10294,10295,10323,10325,10331,10333],{},[411,10296,10297,10321],{},[124,10298,10299,10301,10303,10305,10307,10309,10311,10313,10315,10317,10319],{},[127,10300,65],{},[130,10302,259],{},[127,10304,2746],{},[130,10306,259],{},[127,10308,2754],{},[130,10310,259],{},[127,10312,2754],{},[130,10314,259],{},[127,10316,2746],{},[130,10318,259],{},[127,10320,65],{},[127,10322,2749],{},[130,10324,680],{},[411,10326,10327,10329],{},[127,10328,2749],{},[127,10330,2749],{},[130,10332,680],{},[127,10334,65],{},[151,10336,10337],{"encoding":153},"\\frac{1 + 5 + 10 + 10 + 5 + 1}{32} = \\frac{32}{32} = 1",[108,10339,10341,10454,10537],{"className":10340,"ariaHidden":132},[158],[108,10342,10344,10347,10445,10448,10451],{"className":10343},[162],[108,10345],{"className":10346,"style":1736},[166],[108,10348,10350,10353,10442],{"className":10349},[171],[108,10351],{"className":10352},[443,977],[108,10354,10356],{"className":10355},[411],[108,10357,10359,10434],{"className":10358},[457,458],[108,10360,10362,10431],{"className":10361},[462],[108,10363,10365,10379,10387],{"className":10364,"style":1663},[466],[108,10366,10367,10370],{"style":993},[108,10368],{"className":10369,"style":997},[474],[108,10371,10373],{"className":10372},[479,480,481,482],[108,10374,10376],{"className":10375},[171,482],[108,10377,2749],{"className":10378},[171,482],[108,10380,10381,10384],{"style":1031},[108,10382],{"className":10383,"style":997},[474],[108,10385],{"className":10386,"style":1039},[1038],[108,10388,10389,10392],{"style":1042},[108,10390],{"className":10391,"style":997},[474],[108,10393,10395],{"className":10394},[479,480,481,482],[108,10396,10398,10401,10404,10407,10410,10413,10416,10419,10422,10425,10428],{"className":10397},[171,482],[108,10399,65],{"className":10400},[171,482],[108,10402,259],{"className":10403},[285,482],[108,10405,2746],{"className":10406},[171,482],[108,10408,259],{"className":10409},[285,482],[108,10411,2754],{"className":10412},[171,482],[108,10414,259],{"className":10415},[285,482],[108,10417,2754],{"className":10418},[171,482],[108,10420,259],{"className":10421},[285,482],[108,10423,2746],{"className":10424},[171,482],[108,10426,259],{"className":10427},[285,482],[108,10429,65],{"className":10430},[171,482],[108,10432,510],{"className":10433},[509],[108,10435,10437],{"className":10436},[462],[108,10438,10440],{"className":10439,"style":517},[466],[108,10441],{},[108,10443],{"className":10444},[523,977],[108,10446],{"className":10447,"style":766},[179],[108,10449,680],{"className":10450},[770],[108,10452],{"className":10453,"style":766},[179],[108,10455,10457,10460,10528,10531,10534],{"className":10456},[162],[108,10458],{"className":10459,"style":1736},[166],[108,10461,10463,10466,10525],{"className":10462},[171],[108,10464],{"className":10465},[443,977],[108,10467,10469],{"className":10468},[411],[108,10470,10472,10517],{"className":10471},[457,458],[108,10473,10475,10514],{"className":10474},[462],[108,10476,10478,10492,10500],{"className":10477,"style":1663},[466],[108,10479,10480,10483],{"style":993},[108,10481],{"className":10482,"style":997},[474],[108,10484,10486],{"className":10485},[479,480,481,482],[108,10487,10489],{"className":10488},[171,482],[108,10490,2749],{"className":10491},[171,482],[108,10493,10494,10497],{"style":1031},[108,10495],{"className":10496,"style":997},[474],[108,10498],{"className":10499,"style":1039},[1038],[108,10501,10502,10505],{"style":1042},[108,10503],{"className":10504,"style":997},[474],[108,10506,10508],{"className":10507},[479,480,481,482],[108,10509,10511],{"className":10510},[171,482],[108,10512,2749],{"className":10513},[171,482],[108,10515,510],{"className":10516},[509],[108,10518,10520],{"className":10519},[462],[108,10521,10523],{"className":10522,"style":517},[466],[108,10524],{},[108,10526],{"className":10527},[523,977],[108,10529],{"className":10530,"style":766},[179],[108,10532,680],{"className":10533},[770],[108,10535],{"className":10536,"style":766},[179],[108,10538,10540,10543],{"className":10539},[162],[108,10541],{"className":10542,"style":237},[166],[108,10544,65],{"className":10545},[171],[15,10547,10548,10549,10608,10609,10637,10638,812],{},"Cada número de la fila 5 dividido entre ",[108,10550,10552,10570],{"className":10551},[111],[108,10553,10555],{"className":10554},[115],[117,10556,10557],{"xmlns":119},[121,10558,10559,10567],{},[124,10560,10561],{},[2506,10562,10563,10565],{},[127,10564,675],{},[127,10566,2746],{},[151,10568,10569],{"encoding":153},"2^5",[108,10571,10573],{"className":10572,"ariaHidden":132},[158],[108,10574,10576,10579],{"className":10575},[162],[108,10577],{"className":10578,"style":2812},[166],[108,10580,10582,10585],{"className":10581},[171],[108,10583,675],{"className":10584},[171],[108,10586,10588],{"className":10587},[2534],[108,10589,10591],{"className":10590},[457],[108,10592,10594],{"className":10593},[462],[108,10595,10597],{"className":10596,"style":2812},[466],[108,10598,10599,10602],{"style":2546},[108,10600],{"className":10601,"style":475},[474],[108,10603,10605],{"className":10604},[479,480,481,482],[108,10606,2746],{"className":10607},[171,482]," nos da una probabilidad, y juntas suman ",[108,10610,10612,10625],{"className":10611},[111],[108,10613,10615],{"className":10614},[115],[117,10616,10617],{"xmlns":119},[121,10618,10619,10623],{},[124,10620,10621],{},[127,10622,65],{},[151,10624,65],{"encoding":153},[108,10626,10628],{"className":10627,"ariaHidden":132},[158],[108,10629,10631,10634],{"className":10630},[162],[108,10632],{"className":10633,"style":237},[166],[108,10635,65],{"className":10636},[171],". El triángulo, la combinatoria y el álgebra no son tres historias diferentes: ",[27,10639,10640],{},"son la misma historia contada en tres idiomas distintos",[19,10642,10644],{"id":10643},"reflexión-final","Reflexión final",[15,10646,10647],{},"A lo largo de este artículo hemos seguido un hilo que conecta tres ramas de las matemáticas a través de un mismo objeto:",[6402,10649,10650,10655,10661],{},[60,10651,2265,10652,10654],{},[27,10653,2268],{}," nos dice de cuántas maneras podemos organizar un conjunto de resultados.",[60,10656,10657,10658,10660],{},"El ",[27,10659,6301],{}," del binomio genera esas mismas cantidades como coeficientes de una fórmula.",[60,10662,2265,10663,10665],{},[27,10664,3231],{}," transforma esos números en predicciones sobre el mundo real.",[15,10667,10668,10669,10671],{},"Y el ",[27,10670,29],{}," es el objeto que las une: una tabla visual donde las tres disciplinas convergen.",[15,10673,10674],{},"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.",[15,10676,10677],{},"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.",[15,10679,10680],{},"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.",[15,10682,10683,10684,10689],{},"Pon la teoría en práctica: ",[2765,10685,10688],{"href":10686,"rel":10687},"https:\u002F\u002Fflipthecoin.app\u002Fes\u002Fplay\u002F",[2769],"lanza una moneda online"," y observa la combinatoria en acción.",{"title":90,"searchDepth":10691,"depth":10691,"links":10692},2,[10693,10694,10698,10699,10702,10703,10706],{"id":21,"depth":10691,"text":22},{"id":51,"depth":10691,"text":52,"children":10695},[10696],{"id":382,"depth":10697,"text":383},3,{"id":2261,"depth":10691,"text":2262},{"id":2773,"depth":10691,"text":2774,"children":10700},[10701],{"id":3225,"depth":10697,"text":3226},{"id":5014,"depth":10691,"text":5015},{"id":6294,"depth":10691,"text":6295,"children":10704},[10705],{"id":8552,"depth":10697,"text":8553},{"id":10643,"depth":10691,"text":10644},"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.","md",{"excerpt":10711},{"type":8,"value":10712},[10713,10715],[11,10714,5],{"id":13},[15,10716,17],{},true,"\u002Fblog\u002Fpascals-triangle-and-probability.es",{"title":5,"description":10708},"triangulo-de-tartaglia-combinatoria-y-probabilidad","blog\u002Fpascals-triangle-and-probability.es","qZQyHpms2FZH6Ff6w11iThobO-PzvEaP5Ib4ymGhG00",{"en":10724,"es":10720},"pascals-triangle-combinatorics-and-coin-flips",{"en":10724,"es":10720},1779926067956]