[{"data":1,"prerenderedAt":10648},["ShallowReactive",2],{"blog-post-en-pascals-triangle-combinatorics-and-coin-flips":3,"locale-switcher-blog-en-pascals-triangle-combinatorics-and-coin-flips":10645,"blog-siblings-en-pascals-triangle-combinatorics-and-coin-flips":10647},{"id":4,"title":5,"_locale":6,"body":7,"date":10629,"description":10630,"extension":10631,"meta":10632,"navigation":10639,"path":10640,"published":10639,"seo":10641,"slug":10642,"stem":10643,"__hash__":10644},"blog\u002Fblog\u002Fpascals-triangle-and-probability.en.md","What Do Pascal's Triangle and a Coin Have in Common?","en",{"type":8,"value":9,"toc":10612},"minimal",[10,14,18,23,40,43,49,53,56,80,83,94,104,379,384,391,527,650,813,816,1073,1377,1824,1856,2259,2263,2272,2482,2489,2555,2616,2757,2760,2770,2774,2860,2866,2872,2926,3002,3221,3225,3231,3365,3397,3837,3840,4508,4658,4661,4933,4937,4940,5024,5108,5240,5246,5378,5381,6173,6176,6182,6185,6205,6213,6217,6224,6319,6322,6518,6550,6555,6785,6844,6849,7171,7241,7246,7752,7849,7859,8235,8472,8476,8541,8853,8856,9824,10199,10202,10468,10563,10567,10570,10588,10594,10597,10600,10603],[11,12,5],"h1",{"id":13},"what-do-pascals-triangle-and-a-coin-have-in-common",[15,16,17],"p",{},"There is a triangle of numbers that has fascinated mathematicians across the globe for centuries. Its construction is so simple a child could draw it in a notebook. And yet, hidden within its rows lies a deep connection between three branches of mathematics that, at first glance, seem to have nothing in common: combinatorics, algebra, and probability. The most surprising part? That connection reveals itself through something as ordinary as flipping a coin.",[19,20,22],"h2",{"id":21},"a-bit-of-history","A bit of history",[15,24,25,26,30,31,35,36,39],{},"In the English-speaking world this pattern is known as ",[27,28,29],"strong",{},"Pascal's triangle",", after the French mathematician Blaise Pascal, who published a treatise in 1654 — the ",[32,33,34],"em",{},"Traité du triangle arithmétique"," — where he formalized its properties and, crucially, connected it for the first time to the problems of chance posed to him by the Chevalier de Méré. In many Spanish-speaking countries it is called the ",[27,37,38],{},"triangle of Tartaglia",", after the Italian mathematician Niccolò Fontana Tartaglia (1499–1557), who studied it in the context of solving cubic equations.",[15,41,42],{},"But the real history of this triangle goes much further back. In 11th-century China, the mathematician Jia Xian was already describing the same pattern for computing coefficients of powers. Later, in the 13th century, Yang Hui reproduced it in a diagram that survives to this day. In Persia, Omar Khayyam worked with the same ideas in the 12th century. And in India, Pingala explored similar combinatorial patterns centuries before the common era.",[15,44,45,46],{},"The fact that civilizations so distant in time and space independently discovered the same triangle tells us something important: this is not an arbitrary human invention, but a structure that emerges naturally whenever anyone asks ",[32,47,48],{},"how many ways can I arrange this?",[19,50,52],{"id":51},"how-to-build-the-triangle","How to build the triangle",[15,54,55],{},"The construction rule is elegantly simple:",[57,58,59,67,73],"ol",{},[60,61,62,63,66],"li",{},"Start by placing a ",[27,64,65],{},"1"," at the top.",[60,68,69,70,72],{},"On each new row, place a ",[27,71,65],{}," at each end.",[60,74,75,76,79],{},"Each inner number is the ",[27,77,78],{},"sum of the two numbers"," directly above it.",[15,81,82],{},"The first few rows look like this:",[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],{},"The first row (the lone ",[27,98,65],{}," at the top) is called ",[27,101,102],{},"row 0",". The next one is row 1, and so on. This is not a quirk: numbering from zero will make everything fit neatly once we connect the triangle to the formulas.",[15,105,106,107,211,212,241,242,211,298,326,327,378],{},"Look at row 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],". The ",[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]," in the center comes from adding the two numbers above it: ",[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]," on the left comes from ",[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],". Always the same rule.",[380,381,383],"h3",{"id":382},"binomial-coefficients","Binomial coefficients",[15,385,386,387,390],{},"Every number in the triangle has a formal name: it is a ",[27,388,389],{},"binomial coefficient",". It is written as:",[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],{},"and read \"",[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]," choose ",[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],"\". Its meaning is straightforward: it represents the number of ways to choose ",[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]," elements from a set of ",[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],", regardless of order.",[15,651,652,653,783,784,812],{},"For example, ",[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]," means there are 6 ways to choose 2 items from 4. And indeed, if you look at row 4 of the triangle, position 2 (counting from 0) is precisely ",[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],{},"The formula to compute it is:",[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],{},"Here, the symbol ",[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]," is called ",[27,1112,1113],{},"factorial"," and means multiplying all whole numbers from ",[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]," up to ",[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],". For instance, ",[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],". By convention, ",[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],{},"Let's verify: ",[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],". Correct.",[15,1825,1826,1827,1855],{},"So row ",[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]," of the triangle contains the values:",[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},"combinatorics-the-art-of-counting-without-listing","Combinatorics: the art of counting without listing",[15,2264,2265,2268,2269,812],{},[27,2266,2267],{},"Combinatorics"," is the branch of mathematics that studies how many distinct ways there are to arrange, group, or choose elements. Its power lies in letting us know the number of possibilities ",[32,2270,2271],{},"without having to list them one by one",[15,2273,2274,2275,2278,2279,2282,2283,2311,2312,2385,2386,812],{},"To see why this is useful, consider flipping a coin. Each flip has two possible outcomes: ",[27,2276,2277],{},"heads"," (O) or ",[27,2280,2281],{},"tails"," (X). If we flip the coin once, there are ",[108,2284,2286,2299],{"className":2285},[111],[108,2287,2289],{"className":2288},[115],[117,2290,2291],{"xmlns":119},[121,2292,2293,2297],{},[124,2294,2295],{},[127,2296,675],{},[151,2298,675],{"encoding":153},[108,2300,2302],{"className":2301,"ariaHidden":132},[158],[108,2303,2305,2308],{"className":2304},[162],[108,2306],{"className":2307,"style":237},[166],[108,2309,675],{"className":2310},[171]," outcomes. If we flip it twice, each of the 2 outcomes of the first flip combines with the 2 outcomes of the second, giving ",[108,2313,2315,2337],{"className":2314},[111],[108,2316,2318],{"className":2317},[115],[117,2319,2320],{"xmlns":119},[121,2321,2322,2334],{},[124,2323,2324,2326,2328,2330,2332],{},[127,2325,675],{},[130,2327,1195],{},[127,2329,675],{},[130,2331,680],{},[127,2333,136],{},[151,2335,2336],{"encoding":153},"2 \\times 2 = 4",[108,2338,2340,2358,2376],{"className":2339,"ariaHidden":132},[158],[108,2341,2343,2346,2349,2352,2355],{"className":2342},[162],[108,2344],{"className":2345,"style":274},[166],[108,2347,675],{"className":2348},[171],[108,2350],{"className":2351,"style":281},[179],[108,2353,1195],{"className":2354},[285],[108,2356],{"className":2357,"style":281},[179],[108,2359,2361,2364,2367,2370,2373],{"className":2360},[162],[108,2362],{"className":2363,"style":237},[166],[108,2365,675],{"className":2366},[171],[108,2368],{"className":2369,"style":766},[179],[108,2371,680],{"className":2372},[770],[108,2374],{"className":2375,"style":766},[179],[108,2377,2379,2382],{"className":2378},[162],[108,2380],{"className":2381,"style":237},[166],[108,2383,136],{"className":2384},[171]," outcomes. With three flips, ",[108,2387,2389,2416],{"className":2388},[111],[108,2390,2392],{"className":2391},[115],[117,2393,2394],{"xmlns":119},[121,2395,2396,2413],{},[124,2397,2398,2400,2402,2404,2406,2408,2410],{},[127,2399,675],{},[130,2401,1195],{},[127,2403,675],{},[130,2405,1195],{},[127,2407,675],{},[130,2409,680],{},[127,2411,2412],{},"8",[151,2414,2415],{"encoding":153},"2 \\times 2 \\times 2 = 8",[108,2417,2419,2437,2455,2473],{"className":2418,"ariaHidden":132},[158],[108,2420,2422,2425,2428,2431,2434],{"className":2421},[162],[108,2423],{"className":2424,"style":274},[166],[108,2426,675],{"className":2427},[171],[108,2429],{"className":2430,"style":281},[179],[108,2432,1195],{"className":2433},[285],[108,2435],{"className":2436,"style":281},[179],[108,2438,2440,2443,2446,2449,2452],{"className":2439},[162],[108,2441],{"className":2442,"style":274},[166],[108,2444,675],{"className":2445},[171],[108,2447],{"className":2448,"style":281},[179],[108,2450,1195],{"className":2451},[285],[108,2453],{"className":2454,"style":281},[179],[108,2456,2458,2461,2464,2467,2470],{"className":2457},[162],[108,2459],{"className":2460,"style":237},[166],[108,2462,675],{"className":2463},[171],[108,2465],{"className":2466,"style":766},[179],[108,2468,680],{"className":2469},[770],[108,2471],{"className":2472,"style":766},[179],[108,2474,2476,2479],{"className":2475},[162],[108,2477],{"className":2478,"style":237},[166],[108,2480,2412],{"className":2481},[171],[15,2483,2484,2485,2488],{},"That pattern is expressed with an ",[27,2486,2487],{},"exponent",":",[15,2490,2492],{"className":2491},[394],[108,2493,2495,2514],{"className":2494},[111],[108,2496,2498],{"className":2497},[115],[117,2499,2500],{"xmlns":119},[121,2501,2502,2511],{},[124,2503,2504],{},[2505,2506,2507,2509],"msup",{},[127,2508,675],{},[415,2510,417],{},[151,2512,2513],{"encoding":153},"2^n",[108,2515,2517],{"className":2516,"ariaHidden":132},[158],[108,2518,2520,2524],{"className":2519},[162],[108,2521],{"className":2522,"style":2523},[166],"height:0.6644em;",[108,2525,2527,2530],{"className":2526},[171],[108,2528,675],{"className":2529},[171],[108,2531,2534],{"className":2532},[2533],"msupsub",[108,2535,2537],{"className":2536},[457],[108,2538,2540],{"className":2539},[462],[108,2541,2543],{"className":2542,"style":2523},[466],[108,2544,2546,2549],{"style":2545},"top:-3.063em;margin-right:0.05em;",[108,2547],{"className":2548,"style":475},[474],[108,2550,2552],{"className":2551},[479,480,481,482],[108,2553,417],{"className":2554},[171,489,482],[15,2556,2557,2558,2586,2587,2615],{},"where ",[108,2559,2561,2574],{"className":2560},[111],[108,2562,2564],{"className":2563},[115],[117,2565,2566],{"xmlns":119},[121,2567,2568,2572],{},[124,2569,2570],{},[415,2571,417],{},[151,2573,417],{"encoding":153},[108,2575,2577],{"className":2576,"ariaHidden":132},[158],[108,2578,2580,2583],{"className":2579},[162],[108,2581],{"className":2582,"style":557},[166],[108,2584,417],{"className":2585},[171,489]," is the number of flips. The exponent tells us how many times we multiply the base (",[108,2588,2590,2603],{"className":2589},[111],[108,2591,2593],{"className":2592},[115],[117,2594,2595],{"xmlns":119},[121,2596,2597,2601],{},[124,2598,2599],{},[127,2600,675],{},[151,2602,675],{"encoding":153},[108,2604,2606],{"className":2605,"ariaHidden":132},[158],[108,2607,2609,2612],{"className":2608},[162],[108,2610],{"className":2611,"style":237},[166],[108,2613,675],{"className":2614},[171],") by itself. So:",[2617,2618,2619,2719],"table",{},[2620,2621,2622],"thead",{},[2623,2624,2625,2658],"tr",{},[2626,2627,2629,2630,423],"th",{"align":2628},"center","Flips (",[108,2631,2633,2646],{"className":2632},[111],[108,2634,2636],{"className":2635},[115],[117,2637,2638],{"xmlns":119},[121,2639,2640,2644],{},[124,2641,2642],{},[415,2643,417],{},[151,2645,417],{"encoding":153},[108,2647,2649],{"className":2648,"ariaHidden":132},[158],[108,2650,2652,2655],{"className":2651},[162],[108,2653],{"className":2654,"style":557},[166],[108,2656,417],{"className":2657},[171,489],[2626,2659,2660,2661,423],{"align":2628},"Possible outcomes (",[108,2662,2664,2681],{"className":2663},[111],[108,2665,2667],{"className":2666},[115],[117,2668,2669],{"xmlns":119},[121,2670,2671,2679],{},[124,2672,2673],{},[2505,2674,2675,2677],{},[127,2676,675],{},[415,2678,417],{},[151,2680,2513],{"encoding":153},[108,2682,2684],{"className":2683,"ariaHidden":132},[158],[108,2685,2687,2690],{"className":2686},[162],[108,2688],{"className":2689,"style":2523},[166],[108,2691,2693,2696],{"className":2692},[171],[108,2694,675],{"className":2695},[171],[108,2697,2699],{"className":2698},[2533],[108,2700,2702],{"className":2701},[457],[108,2703,2705],{"className":2704},[462],[108,2706,2708],{"className":2707,"style":2523},[466],[108,2709,2710,2713],{"style":2545},[108,2711],{"className":2712,"style":475},[474],[108,2714,2716],{"className":2715},[479,480,481,482],[108,2717,417],{"className":2718},[171,489,482],[2720,2721,2722,2729,2735,2741,2749],"tbody",{},[2623,2723,2724,2727],{},[2725,2726,65],"td",{"align":2628},[2725,2728,675],{"align":2628},[2623,2730,2731,2733],{},[2725,2732,675],{"align":2628},[2725,2734,136],{"align":2628},[2623,2736,2737,2739],{},[2725,2738,256],{"align":2628},[2725,2740,2412],{"align":2628},[2623,2742,2743,2746],{},[2725,2744,2745],{"align":2628},"5",[2725,2747,2748],{"align":2628},"32",[2623,2750,2751,2754],{},[2725,2752,2753],{"align":2628},"10",[2725,2755,2756],{"align":2628},"1,024",[15,2758,2759],{},"With just 10 flips there are already over a thousand possible combinations. Combinatorics helps us handle those numbers without losing our minds.",[15,2761,2762,2763,812],{},"If you need a refresher on what probability means and how it is measured, check out ",[2764,2765,2769],"a",{"href":2766,"rel":2767},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fprobability-basics\u002F",[2768],"nofollow","Probability Basics",[19,2771,2773],{"id":2772},"case-study-3-flips","Case study: 3 flips",[15,2775,2776,2777,2859],{},"Let's connect everything with a concrete example. We flip a coin 3 times. We know there are ",[108,2778,2780,2802],{"className":2779},[111],[108,2781,2783],{"className":2782},[115],[117,2784,2785],{"xmlns":119},[121,2786,2787,2799],{},[124,2788,2789,2795,2797],{},[2505,2790,2791,2793],{},[127,2792,675],{},[127,2794,256],{},[130,2796,680],{},[127,2798,2412],{},[151,2800,2801],{"encoding":153},"2^3 = 8",[108,2803,2805,2850],{"className":2804,"ariaHidden":132},[158],[108,2806,2808,2812,2841,2844,2847],{"className":2807},[162],[108,2809],{"className":2810,"style":2811},[166],"height:0.8141em;",[108,2813,2815,2818],{"className":2814},[171],[108,2816,675],{"className":2817},[171],[108,2819,2821],{"className":2820},[2533],[108,2822,2824],{"className":2823},[457],[108,2825,2827],{"className":2826},[462],[108,2828,2830],{"className":2829,"style":2811},[466],[108,2831,2832,2835],{"style":2545},[108,2833],{"className":2834,"style":475},[474],[108,2836,2838],{"className":2837},[479,480,481,482],[108,2839,256],{"className":2840},[171,482],[108,2842],{"className":2843,"style":766},[179],[108,2845,680],{"className":2846},[770],[108,2848],{"className":2849,"style":766},[179],[108,2851,2853,2856],{"className":2852},[162],[108,2854],{"className":2855,"style":237},[166],[108,2857,2412],{"className":2858},[171]," possible outcomes. Here they all are:",[84,2861,2864],{"className":2862,"code":2863,"language":89,"meta":90},[87],"OOO    OOX    OXO    XOO    OXX    XOX    XXO    XXX\n",[92,2865,2863],{"__ignoreMap":90},[15,2867,2868,2869,2488],{},"Now let's group these results by the ",[27,2870,2871],{},"number of heads",[2617,2873,2874,2888],{},[2620,2875,2876],{},[2623,2877,2878,2881,2885],{},[2626,2879,2880],{"align":2628},"Heads",[2626,2882,2884],{"align":2883},"left","Combinations",[2626,2886,2887],{"align":2628},"Count",[2720,2889,2890,2899,2908,2917],{},[2623,2891,2892,2894,2897],{},[2725,2893,1334],{"align":2628},[2725,2895,2896],{"align":2883},"XXX",[2725,2898,65],{"align":2628},[2623,2900,2901,2903,2906],{},[2725,2902,65],{"align":2628},[2725,2904,2905],{"align":2883},"OXX, XOX, XXO",[2725,2907,256],{"align":2628},[2623,2909,2910,2912,2915],{},[2725,2911,675],{"align":2628},[2725,2913,2914],{"align":2883},"OOX, OXO, XOO",[2725,2916,256],{"align":2628},[2623,2918,2919,2921,2924],{},[2725,2920,256],{"align":2628},[2725,2922,2923],{"align":2883},"OOO",[2725,2925,65],{"align":2628},[15,2927,2928,2929,2997,2998,3001],{},"Look at the count column: ",[108,2930,2932,2958],{"className":2931},[111],[108,2933,2935],{"className":2934},[115],[117,2936,2937],{"xmlns":119},[121,2938,2939,2955],{},[124,2940,2941,2943,2945,2947,2949,2951,2953],{},[127,2942,65],{},[130,2944,133],{"separator":132},[127,2946,256],{},[130,2948,133],{"separator":132},[127,2950,256],{},[130,2952,133],{"separator":132},[127,2954,65],{},[151,2956,2957],{"encoding":153},"1, 3, 3, 1",[108,2959,2961],{"className":2960,"ariaHidden":132},[158],[108,2962,2964,2967,2970,2973,2976,2979,2982,2985,2988,2991,2994],{"className":2963},[162],[108,2965],{"className":2966,"style":167},[166],[108,2968,65],{"className":2969},[171],[108,2971,133],{"className":2972},[175],[108,2974],{"className":2975,"style":180},[179],[108,2977,256],{"className":2978},[171],[108,2980,133],{"className":2981},[175],[108,2983],{"className":2984,"style":180},[179],[108,2986,256],{"className":2987},[171],[108,2989,133],{"className":2990},[175],[108,2992],{"className":2993,"style":180},[179],[108,2995,65],{"className":2996},[171],". Sound familiar? It is exactly ",[27,2999,3000],{},"row 3"," of Pascal's triangle.",[15,3003,3004,3005,3033,3034,3062,3063,3162,3163,3191,3192,3220],{},"And it is no coincidence. The number of ways to get exactly ",[108,3006,3008,3021],{"className":3007},[111],[108,3009,3011],{"className":3010},[115],[117,3012,3013],{"xmlns":119},[121,3014,3015,3019],{},[124,3016,3017],{},[415,3018,420],{},[151,3020,420],{"encoding":153},[108,3022,3024],{"className":3023,"ariaHidden":132},[158],[108,3025,3027,3030],{"className":3026},[162],[108,3028],{"className":3029,"style":587},[166],[108,3031,420],{"className":3032,"style":490},[171,489]," heads in ",[108,3035,3037,3050],{"className":3036},[111],[108,3038,3040],{"className":3039},[115],[117,3041,3042],{"xmlns":119},[121,3043,3044,3048],{},[124,3045,3046],{},[415,3047,417],{},[151,3049,417],{"encoding":153},[108,3051,3053],{"className":3052,"ariaHidden":132},[158],[108,3054,3056,3059],{"className":3055},[162],[108,3057],{"className":3058,"style":557},[166],[108,3060,417],{"className":3061},[171,489]," flips is precisely ",[108,3064,3066,3087],{"className":3065},[111],[108,3067,3069],{"className":3068},[115],[117,3070,3071],{"xmlns":119},[121,3072,3073,3085],{},[124,3074,3075,3077,3083],{},[130,3076,409],{"fence":132},[411,3078,3079,3081],{"linethickness":413},[415,3080,417],{},[415,3082,420],{},[130,3084,423],{"fence":132},[151,3086,426],{"encoding":153},[108,3088,3090],{"className":3089,"ariaHidden":132},[158],[108,3091,3093,3096],{"className":3092},[162],[108,3094],{"className":3095,"style":436},[166],[108,3097,3099,3105,3156],{"className":3098},[171],[108,3100,3102],{"className":3101,"style":445},[443,444],[108,3103,409],{"className":3104},[449,450],[108,3106,3108],{"className":3107},[411],[108,3109,3111,3148],{"className":3110},[457,458],[108,3112,3114,3145],{"className":3113},[462],[108,3115,3117,3131],{"className":3116,"style":467},[466],[108,3118,3119,3122],{"style":470},[108,3120],{"className":3121,"style":475},[474],[108,3123,3125],{"className":3124},[479,480,481,482],[108,3126,3128],{"className":3127},[171,482],[108,3129,420],{"className":3130,"style":490},[171,489,482],[108,3132,3133,3136],{"style":493},[108,3134],{"className":3135,"style":475},[474],[108,3137,3139],{"className":3138},[479,480,481,482],[108,3140,3142],{"className":3141},[171,482],[108,3143,417],{"className":3144},[171,489,482],[108,3146,510],{"className":3147},[509],[108,3149,3151],{"className":3150},[462],[108,3152,3154],{"className":3153,"style":517},[466],[108,3155],{},[108,3157,3159],{"className":3158,"style":445},[523,444],[108,3160,423],{"className":3161},[449,450]," — the binomial coefficient at position ",[108,3164,3166,3179],{"className":3165},[111],[108,3167,3169],{"className":3168},[115],[117,3170,3171],{"xmlns":119},[121,3172,3173,3177],{},[124,3174,3175],{},[415,3176,420],{},[151,3178,420],{"encoding":153},[108,3180,3182],{"className":3181,"ariaHidden":132},[158],[108,3183,3185,3188],{"className":3184},[162],[108,3186],{"className":3187,"style":587},[166],[108,3189,420],{"className":3190,"style":490},[171,489]," in row ",[108,3193,3195,3208],{"className":3194},[111],[108,3196,3198],{"className":3197},[115],[117,3199,3200],{"xmlns":119},[121,3201,3202,3206],{},[124,3203,3204],{},[415,3205,417],{},[151,3207,417],{"encoding":153},[108,3209,3211],{"className":3210,"ariaHidden":132},[158],[108,3212,3214,3217],{"className":3213},[162],[108,3215],{"className":3216,"style":557},[166],[108,3218,417],{"className":3219},[171,489]," of the triangle.",[380,3222,3224],{"id":3223},"probability-enters-the-scene","Probability enters the scene",[15,3226,3227,3230],{},[27,3228,3229],{},"Probability"," measures how likely an event is to occur. It is calculated as:",[15,3232,3234],{"className":3233},[394],[108,3235,3237,3262],{"className":3236},[111],[108,3238,3240],{"className":3239},[115],[117,3241,3242],{"xmlns":119},[121,3243,3244,3259],{},[124,3245,3246,3249,3251],{},[415,3247,3248],{},"P",[130,3250,680],{},[411,3252,3253,3256],{},[1886,3254,3255],{},"favorable outcomes",[1886,3257,3258],{},"total outcomes",[151,3260,3261],{"encoding":153},"P = \\frac{\\text{favorable outcomes}}{\\text{total outcomes}}",[108,3263,3265,3285],{"className":3264,"ariaHidden":132},[158],[108,3266,3268,3272,3276,3279,3282],{"className":3267},[162],[108,3269],{"className":3270,"style":3271},[166],"height:0.6833em;",[108,3273,3248],{"className":3274,"style":3275},[171,489],"margin-right:0.1389em;",[108,3277],{"className":3278,"style":766},[179],[108,3280,680],{"className":3281},[770],[108,3283],{"className":3284,"style":766},[179],[108,3286,3288,3291],{"className":3287},[162],[108,3289],{"className":3290,"style":1545},[166],[108,3292,3294,3297,3362],{"className":3293},[171],[108,3295],{"className":3296},[443,977],[108,3298,3300],{"className":3299},[411],[108,3301,3303,3354],{"className":3302},[457,458],[108,3304,3306,3351],{"className":3305},[462],[108,3307,3309,3326,3334],{"className":3308,"style":990},[466],[108,3310,3311,3314],{"style":993},[108,3312],{"className":3313,"style":997},[474],[108,3315,3317],{"className":3316},[479,480,481,482],[108,3318,3320],{"className":3319},[171,482],[108,3321,3323],{"className":3322},[171,89,482],[108,3324,3258],{"className":3325},[171,482],[108,3327,3328,3331],{"style":1031},[108,3329],{"className":3330,"style":997},[474],[108,3332],{"className":3333,"style":1039},[1038],[108,3335,3336,3339],{"style":1042},[108,3337],{"className":3338,"style":997},[474],[108,3340,3342],{"className":3341},[479,480,481,482],[108,3343,3345],{"className":3344},[171,482],[108,3346,3348],{"className":3347},[171,89,482],[108,3349,3255],{"className":3350},[171,482],[108,3352,510],{"className":3353},[509],[108,3355,3357],{"className":3356},[462],[108,3358,3360],{"className":3359,"style":517},[466],[108,3361],{},[108,3363],{"className":3364},[523,977],[15,3366,3367,3368,3396],{},"If the coin is fair, each of the 8 outcomes is equally likely. So the probability of getting exactly ",[108,3369,3371,3384],{"className":3370},[111],[108,3372,3374],{"className":3373},[115],[117,3375,3376],{"xmlns":119},[121,3377,3378,3382],{},[124,3379,3380],{},[415,3381,420],{},[151,3383,420],{"encoding":153},[108,3385,3387],{"className":3386,"ariaHidden":132},[158],[108,3388,3390,3393],{"className":3389},[162],[108,3391],{"className":3392,"style":587},[166],[108,3394,420],{"className":3395,"style":490},[171,489]," heads in 3 flips is:",[15,3398,3400],{"className":3399},[394],[108,3401,3403,3466],{"className":3402},[111],[108,3404,3406],{"className":3405},[115],[117,3407,3408],{"xmlns":119},[121,3409,3410,3463],{},[124,3411,3412,3414,3416,3418,3421,3423,3425,3445,3447],{},[415,3413,3248],{},[130,3415,409],{"stretchy":853},[415,3417,420],{},[1886,3419,3420],{}," heads",[130,3422,423],{"stretchy":853},[130,3424,680],{},[411,3426,3427,3439],{},[124,3428,3429,3431,3437],{},[130,3430,409],{"fence":132},[411,3432,3433,3435],{"linethickness":413},[127,3434,256],{},[415,3436,420],{},[130,3438,423],{"fence":132},[2505,3440,3441,3443],{},[127,3442,675],{},[127,3444,256],{},[130,3446,680],{},[411,3448,3449,3461],{},[124,3450,3451,3453,3459],{},[130,3452,409],{"fence":132},[411,3454,3455,3457],{"linethickness":413},[127,3456,256],{},[415,3458,420],{},[130,3460,423],{"fence":132},[127,3462,2412],{},[151,3464,3465],{"encoding":153},"P(k \\text{ heads}) = \\frac{\\binom{3}{k}}{2^3} = \\frac{\\binom{3}{k}}{8}",[108,3467,3469,3503,3694],{"className":3468,"ariaHidden":132},[158],[108,3470,3472,3476,3479,3482,3485,3491,3494,3497,3500],{"className":3471},[162],[108,3473],{"className":3474,"style":3475},[166],"height:1em;vertical-align:-0.25em;",[108,3477,3248],{"className":3478,"style":3275},[171,489],[108,3480,409],{"className":3481},[443],[108,3483,420],{"className":3484,"style":490},[171,489],[108,3486,3488],{"className":3487},[171,89],[108,3489,3420],{"className":3490},[171],[108,3492,423],{"className":3493},[523],[108,3495],{"className":3496,"style":766},[179],[108,3498,680],{"className":3499},[770],[108,3501],{"className":3502,"style":766},[179],[108,3504,3506,3510,3685,3688,3691],{"className":3505},[162],[108,3507],{"className":3508,"style":3509},[166],"height:1.855em;vertical-align:-0.345em;",[108,3511,3513,3516,3682],{"className":3512},[171],[108,3514],{"className":3515},[443,977],[108,3517,3519],{"className":3518},[411],[108,3520,3522,3674],{"className":3521},[457,458],[108,3523,3525,3671],{"className":3524},[462],[108,3526,3529,3573,3581],{"className":3527,"style":3528},[466],"height:1.51em;",[108,3530,3531,3534],{"style":993},[108,3532],{"className":3533,"style":997},[474],[108,3535,3537],{"className":3536},[479,480,481,482],[108,3538,3540],{"className":3539},[171,482],[108,3541,3543,3546],{"className":3542},[171,482],[108,3544,675],{"className":3545},[171,482],[108,3547,3549],{"className":3548},[2533],[108,3550,3552],{"className":3551},[457],[108,3553,3555],{"className":3554},[462],[108,3556,3559],{"className":3557,"style":3558},[466],"height:0.7463em;",[108,3560,3562,3566],{"style":3561},"top:-2.786em;margin-right:0.0714em;",[108,3563],{"className":3564,"style":3565},[474],"height:2.5em;",[108,3567,3570],{"className":3568},[479,3569,450,482],"reset-size3",[108,3571,256],{"className":3572},[171,482],[108,3574,3575,3578],{"style":1031},[108,3576],{"className":3577,"style":997},[474],[108,3579],{"className":3580,"style":1039},[1038],[108,3582,3584,3587],{"style":3583},"top:-3.7125em;",[108,3585],{"className":3586,"style":997},[474],[108,3588,3590],{"className":3589},[479,480,481,482],[108,3591,3593],{"className":3592},[171,482],[108,3594,3596,3607,3662],{"className":3595},[171,482],[108,3597,3601],{"className":3598,"style":3600},[443,479,3569,3599,482,444],"size6","top:0.075em;",[108,3602,3604],{"className":3603},[449,450,482],[108,3605,409],{"className":3606},[482],[108,3608,3610],{"className":3609},[411],[108,3611,3613,3653],{"className":3612},[457,458],[108,3614,3616,3650],{"className":3615},[462],[108,3617,3620,3635],{"className":3618,"style":3619},[466],"height:0.9313em;",[108,3621,3623,3626],{"style":3622},"top:-2.156em;",[108,3624],{"className":3625,"style":3565},[474],[108,3627,3629],{"className":3628},[479,3569,450,482],[108,3630,3632],{"className":3631},[171,482],[108,3633,420],{"className":3634,"style":490},[171,489,482],[108,3636,3638,3641],{"style":3637},"top:-2.971em;",[108,3639],{"className":3640,"style":3565},[474],[108,3642,3644],{"className":3643},[479,3569,450,482],[108,3645,3647],{"className":3646},[171,482],[108,3648,256],{"className":3649},[171,482],[108,3651,510],{"className":3652},[509],[108,3654,3656],{"className":3655},[462],[108,3657,3660],{"className":3658,"style":3659},[466],"height:0.344em;",[108,3661],{},[108,3663,3665],{"className":3664,"style":3600},[523,479,3569,3599,482,444],[108,3666,3668],{"className":3667},[449,450,482],[108,3669,423],{"className":3670},[482],[108,3672,510],{"className":3673},[509],[108,3675,3677],{"className":3676},[462],[108,3678,3680],{"className":3679,"style":517},[466],[108,3681],{},[108,3683],{"className":3684},[523,977],[108,3686],{"className":3687,"style":766},[179],[108,3689,680],{"className":3690},[770],[108,3692],{"className":3693,"style":766},[179],[108,3695,3697,3700],{"className":3696},[162],[108,3698],{"className":3699,"style":3509},[166],[108,3701,3703,3706,3834],{"className":3702},[171],[108,3704],{"className":3705},[443,977],[108,3707,3709],{"className":3708},[411],[108,3710,3712,3826],{"className":3711},[457,458],[108,3713,3715,3823],{"className":3714},[462],[108,3716,3718,3732,3740],{"className":3717,"style":3528},[466],[108,3719,3720,3723],{"style":993},[108,3721],{"className":3722,"style":997},[474],[108,3724,3726],{"className":3725},[479,480,481,482],[108,3727,3729],{"className":3728},[171,482],[108,3730,2412],{"className":3731},[171,482],[108,3733,3734,3737],{"style":1031},[108,3735],{"className":3736,"style":997},[474],[108,3738],{"className":3739,"style":1039},[1038],[108,3741,3742,3745],{"style":3583},[108,3743],{"className":3744,"style":997},[474],[108,3746,3748],{"className":3747},[479,480,481,482],[108,3749,3751],{"className":3750},[171,482],[108,3752,3754,3763,3814],{"className":3753},[171,482],[108,3755,3757],{"className":3756,"style":3600},[443,479,3569,3599,482,444],[108,3758,3760],{"className":3759},[449,450,482],[108,3761,409],{"className":3762},[482],[108,3764,3766],{"className":3765},[411],[108,3767,3769,3806],{"className":3768},[457,458],[108,3770,3772,3803],{"className":3771},[462],[108,3773,3775,3789],{"className":3774,"style":3619},[466],[108,3776,3777,3780],{"style":3622},[108,3778],{"className":3779,"style":3565},[474],[108,3781,3783],{"className":3782},[479,3569,450,482],[108,3784,3786],{"className":3785},[171,482],[108,3787,420],{"className":3788,"style":490},[171,489,482],[108,3790,3791,3794],{"style":3637},[108,3792],{"className":3793,"style":3565},[474],[108,3795,3797],{"className":3796},[479,3569,450,482],[108,3798,3800],{"className":3799},[171,482],[108,3801,256],{"className":3802},[171,482],[108,3804,510],{"className":3805},[509],[108,3807,3809],{"className":3808},[462],[108,3810,3812],{"className":3811,"style":3659},[466],[108,3813],{},[108,3815,3817],{"className":3816,"style":3600},[523,479,3569,3599,482,444],[108,3818,3820],{"className":3819},[449,450,482],[108,3821,423],{"className":3822},[482],[108,3824,510],{"className":3825},[509],[108,3827,3829],{"className":3828},[462],[108,3830,3832],{"className":3831,"style":517},[466],[108,3833],{},[108,3835],{"className":3836},[523,977],[15,3838,3839],{},"Let's compute each case:",[2617,3841,3842,3984],{},[2620,3843,3844],{},[2623,3845,3846,3877,3979,3981],{},[2626,3847,3848,3876],{"align":2628},[108,3849,3851,3864],{"className":3850},[111],[108,3852,3854],{"className":3853},[115],[117,3855,3856],{"xmlns":119},[121,3857,3858,3862],{},[124,3859,3860],{},[415,3861,420],{},[151,3863,420],{"encoding":153},[108,3865,3867],{"className":3866,"ariaHidden":132},[158],[108,3868,3870,3873],{"className":3869},[162],[108,3871],{"className":3872,"style":587},[166],[108,3874,420],{"className":3875,"style":490},[171,489]," heads",[2626,3878,3879],{"align":2628},[108,3880,3882,3904],{"className":3881},[111],[108,3883,3885],{"className":3884},[115],[117,3886,3887],{"xmlns":119},[121,3888,3889,3901],{},[124,3890,3891,3893,3899],{},[130,3892,409],{"fence":132},[411,3894,3895,3897],{"linethickness":413},[127,3896,256],{},[415,3898,420],{},[130,3900,423],{"fence":132},[151,3902,3903],{"encoding":153},"\\binom{3}{k}",[108,3905,3907],{"className":3906,"ariaHidden":132},[158],[108,3908,3910,3913],{"className":3909},[162],[108,3911],{"className":3912,"style":695},[166],[108,3914,3916,3922,3973],{"className":3915},[171],[108,3917,3919],{"className":3918,"style":445},[443,444],[108,3920,409],{"className":3921},[449,450],[108,3923,3925],{"className":3924},[411],[108,3926,3928,3965],{"className":3927},[457,458],[108,3929,3931,3962],{"className":3930},[462],[108,3932,3934,3948],{"className":3933,"style":717},[466],[108,3935,3936,3939],{"style":470},[108,3937],{"className":3938,"style":475},[474],[108,3940,3942],{"className":3941},[479,480,481,482],[108,3943,3945],{"className":3944},[171,482],[108,3946,420],{"className":3947,"style":490},[171,489,482],[108,3949,3950,3953],{"style":493},[108,3951],{"className":3952,"style":475},[474],[108,3954,3956],{"className":3955},[479,480,481,482],[108,3957,3959],{"className":3958},[171,482],[108,3960,256],{"className":3961},[171,482],[108,3963,510],{"className":3964},[509],[108,3966,3968],{"className":3967},[462],[108,3969,3971],{"className":3970,"style":517},[466],[108,3972],{},[108,3974,3976],{"className":3975,"style":445},[523,444],[108,3977,423],{"className":3978},[449,450],[2626,3980,3229],{"align":2628},[2626,3982,3983],{"align":2628},"Percentage",[2720,3985,3986,4118,4250,4379],{},[2623,3987,3988,3990,3992,4115],{},[2725,3989,1334],{"align":2628},[2725,3991,65],{"align":2628},[2725,3993,3994],{"align":2628},[108,3995,3997,4020],{"className":3996},[111],[108,3998,4000],{"className":3999},[115],[117,4001,4002],{"xmlns":119},[121,4003,4004,4017],{},[124,4005,4006,4012,4014],{},[411,4007,4008,4010],{},[127,4009,65],{},[127,4011,2412],{},[130,4013,680],{},[127,4015,4016],{},"0.125",[151,4018,4019],{"encoding":153},"\\frac{1}{8} = 0.125",[108,4021,4023,4106],{"className":4022,"ariaHidden":132},[158],[108,4024,4026,4029,4097,4100,4103],{"className":4025},[162],[108,4027],{"className":4028,"style":1736},[166],[108,4030,4032,4035,4094],{"className":4031},[171],[108,4033],{"className":4034},[443,977],[108,4036,4038],{"className":4037},[411],[108,4039,4041,4086],{"className":4040},[457,458],[108,4042,4044,4083],{"className":4043},[462],[108,4045,4047,4061,4069],{"className":4046,"style":1663},[466],[108,4048,4049,4052],{"style":993},[108,4050],{"className":4051,"style":997},[474],[108,4053,4055],{"className":4054},[479,480,481,482],[108,4056,4058],{"className":4057},[171,482],[108,4059,2412],{"className":4060},[171,482],[108,4062,4063,4066],{"style":1031},[108,4064],{"className":4065,"style":997},[474],[108,4067],{"className":4068,"style":1039},[1038],[108,4070,4071,4074],{"style":1042},[108,4072],{"className":4073,"style":997},[474],[108,4075,4077],{"className":4076},[479,480,481,482],[108,4078,4080],{"className":4079},[171,482],[108,4081,65],{"className":4082},[171,482],[108,4084,510],{"className":4085},[509],[108,4087,4089],{"className":4088},[462],[108,4090,4092],{"className":4091,"style":517},[466],[108,4093],{},[108,4095],{"className":4096},[523,977],[108,4098],{"className":4099,"style":766},[179],[108,4101,680],{"className":4102},[770],[108,4104],{"className":4105,"style":766},[179],[108,4107,4109,4112],{"className":4108},[162],[108,4110],{"className":4111,"style":237},[166],[108,4113,4016],{"className":4114},[171],[2725,4116,4117],{"align":2628},"12.5%",[2623,4119,4120,4122,4124,4247],{},[2725,4121,65],{"align":2628},[2725,4123,256],{"align":2628},[2725,4125,4126],{"align":2628},[108,4127,4129,4152],{"className":4128},[111],[108,4130,4132],{"className":4131},[115],[117,4133,4134],{"xmlns":119},[121,4135,4136,4149],{},[124,4137,4138,4144,4146],{},[411,4139,4140,4142],{},[127,4141,256],{},[127,4143,2412],{},[130,4145,680],{},[127,4147,4148],{},"0.375",[151,4150,4151],{"encoding":153},"\\frac{3}{8} = 0.375",[108,4153,4155,4238],{"className":4154,"ariaHidden":132},[158],[108,4156,4158,4161,4229,4232,4235],{"className":4157},[162],[108,4159],{"className":4160,"style":1736},[166],[108,4162,4164,4167,4226],{"className":4163},[171],[108,4165],{"className":4166},[443,977],[108,4168,4170],{"className":4169},[411],[108,4171,4173,4218],{"className":4172},[457,458],[108,4174,4176,4215],{"className":4175},[462],[108,4177,4179,4193,4201],{"className":4178,"style":1663},[466],[108,4180,4181,4184],{"style":993},[108,4182],{"className":4183,"style":997},[474],[108,4185,4187],{"className":4186},[479,480,481,482],[108,4188,4190],{"className":4189},[171,482],[108,4191,2412],{"className":4192},[171,482],[108,4194,4195,4198],{"style":1031},[108,4196],{"className":4197,"style":997},[474],[108,4199],{"className":4200,"style":1039},[1038],[108,4202,4203,4206],{"style":1042},[108,4204],{"className":4205,"style":997},[474],[108,4207,4209],{"className":4208},[479,480,481,482],[108,4210,4212],{"className":4211},[171,482],[108,4213,256],{"className":4214},[171,482],[108,4216,510],{"className":4217},[509],[108,4219,4221],{"className":4220},[462],[108,4222,4224],{"className":4223,"style":517},[466],[108,4225],{},[108,4227],{"className":4228},[523,977],[108,4230],{"className":4231,"style":766},[179],[108,4233,680],{"className":4234},[770],[108,4236],{"className":4237,"style":766},[179],[108,4239,4241,4244],{"className":4240},[162],[108,4242],{"className":4243,"style":237},[166],[108,4245,4148],{"className":4246},[171],[2725,4248,4249],{"align":2628},"37.5%",[2623,4251,4252,4254,4256,4377],{},[2725,4253,675],{"align":2628},[2725,4255,256],{"align":2628},[2725,4257,4258],{"align":2628},[108,4259,4261,4282],{"className":4260},[111],[108,4262,4264],{"className":4263},[115],[117,4265,4266],{"xmlns":119},[121,4267,4268,4280],{},[124,4269,4270,4276,4278],{},[411,4271,4272,4274],{},[127,4273,256],{},[127,4275,2412],{},[130,4277,680],{},[127,4279,4148],{},[151,4281,4151],{"encoding":153},[108,4283,4285,4368],{"className":4284,"ariaHidden":132},[158],[108,4286,4288,4291,4359,4362,4365],{"className":4287},[162],[108,4289],{"className":4290,"style":1736},[166],[108,4292,4294,4297,4356],{"className":4293},[171],[108,4295],{"className":4296},[443,977],[108,4298,4300],{"className":4299},[411],[108,4301,4303,4348],{"className":4302},[457,458],[108,4304,4306,4345],{"className":4305},[462],[108,4307,4309,4323,4331],{"className":4308,"style":1663},[466],[108,4310,4311,4314],{"style":993},[108,4312],{"className":4313,"style":997},[474],[108,4315,4317],{"className":4316},[479,480,481,482],[108,4318,4320],{"className":4319},[171,482],[108,4321,2412],{"className":4322},[171,482],[108,4324,4325,4328],{"style":1031},[108,4326],{"className":4327,"style":997},[474],[108,4329],{"className":4330,"style":1039},[1038],[108,4332,4333,4336],{"style":1042},[108,4334],{"className":4335,"style":997},[474],[108,4337,4339],{"className":4338},[479,480,481,482],[108,4340,4342],{"className":4341},[171,482],[108,4343,256],{"className":4344},[171,482],[108,4346,510],{"className":4347},[509],[108,4349,4351],{"className":4350},[462],[108,4352,4354],{"className":4353,"style":517},[466],[108,4355],{},[108,4357],{"className":4358},[523,977],[108,4360],{"className":4361,"style":766},[179],[108,4363,680],{"className":4364},[770],[108,4366],{"className":4367,"style":766},[179],[108,4369,4371,4374],{"className":4370},[162],[108,4372],{"className":4373,"style":237},[166],[108,4375,4148],{"className":4376},[171],[2725,4378,4249],{"align":2628},[2623,4380,4381,4383,4385,4506],{},[2725,4382,256],{"align":2628},[2725,4384,65],{"align":2628},[2725,4386,4387],{"align":2628},[108,4388,4390,4411],{"className":4389},[111],[108,4391,4393],{"className":4392},[115],[117,4394,4395],{"xmlns":119},[121,4396,4397,4409],{},[124,4398,4399,4405,4407],{},[411,4400,4401,4403],{},[127,4402,65],{},[127,4404,2412],{},[130,4406,680],{},[127,4408,4016],{},[151,4410,4019],{"encoding":153},[108,4412,4414,4497],{"className":4413,"ariaHidden":132},[158],[108,4415,4417,4420,4488,4491,4494],{"className":4416},[162],[108,4418],{"className":4419,"style":1736},[166],[108,4421,4423,4426,4485],{"className":4422},[171],[108,4424],{"className":4425},[443,977],[108,4427,4429],{"className":4428},[411],[108,4430,4432,4477],{"className":4431},[457,458],[108,4433,4435,4474],{"className":4434},[462],[108,4436,4438,4452,4460],{"className":4437,"style":1663},[466],[108,4439,4440,4443],{"style":993},[108,4441],{"className":4442,"style":997},[474],[108,4444,4446],{"className":4445},[479,480,481,482],[108,4447,4449],{"className":4448},[171,482],[108,4450,2412],{"className":4451},[171,482],[108,4453,4454,4457],{"style":1031},[108,4455],{"className":4456,"style":997},[474],[108,4458],{"className":4459,"style":1039},[1038],[108,4461,4462,4465],{"style":1042},[108,4463],{"className":4464,"style":997},[474],[108,4466,4468],{"className":4467},[479,480,481,482],[108,4469,4471],{"className":4470},[171,482],[108,4472,65],{"className":4473},[171,482],[108,4475,510],{"className":4476},[509],[108,4478,4480],{"className":4479},[462],[108,4481,4483],{"className":4482,"style":517},[466],[108,4484],{},[108,4486],{"className":4487},[523,977],[108,4489],{"className":4490,"style":766},[179],[108,4492,680],{"className":4493},[770],[108,4495],{"className":4496,"style":766},[179],[108,4498,4500,4503],{"className":4499},[162],[108,4501],{"className":4502,"style":237},[166],[108,4504,4016],{"className":4505},[171],[2725,4507,4117],{"align":2628},[15,4509,4510,4511,4628,4629,4657],{},"If we add all probabilities: ",[108,4512,4514,4544],{"className":4513},[111],[108,4515,4517],{"className":4516},[115],[117,4518,4519],{"xmlns":119},[121,4520,4521,4541],{},[124,4522,4523,4525,4527,4529,4531,4533,4535,4537,4539],{},[127,4524,4016],{},[130,4526,259],{},[127,4528,4148],{},[130,4530,259],{},[127,4532,4148],{},[130,4534,259],{},[127,4536,4016],{},[130,4538,680],{},[127,4540,65],{},[151,4542,4543],{"encoding":153},"0.125 + 0.375 + 0.375 + 0.125 = 1",[108,4545,4547,4565,4583,4601,4619],{"className":4546,"ariaHidden":132},[158],[108,4548,4550,4553,4556,4559,4562],{"className":4549},[162],[108,4551],{"className":4552,"style":274},[166],[108,4554,4016],{"className":4555},[171],[108,4557],{"className":4558,"style":281},[179],[108,4560,259],{"className":4561},[285],[108,4563],{"className":4564,"style":281},[179],[108,4566,4568,4571,4574,4577,4580],{"className":4567},[162],[108,4569],{"className":4570,"style":274},[166],[108,4572,4148],{"className":4573},[171],[108,4575],{"className":4576,"style":281},[179],[108,4578,259],{"className":4579},[285],[108,4581],{"className":4582,"style":281},[179],[108,4584,4586,4589,4592,4595,4598],{"className":4585},[162],[108,4587],{"className":4588,"style":274},[166],[108,4590,4148],{"className":4591},[171],[108,4593],{"className":4594,"style":281},[179],[108,4596,259],{"className":4597},[285],[108,4599],{"className":4600,"style":281},[179],[108,4602,4604,4607,4610,4613,4616],{"className":4603},[162],[108,4605],{"className":4606,"style":237},[166],[108,4608,4016],{"className":4609},[171],[108,4611],{"className":4612,"style":766},[179],[108,4614,680],{"className":4615},[770],[108,4617],{"className":4618,"style":766},[179],[108,4620,4622,4625],{"className":4621},[162],[108,4623],{"className":4624,"style":237},[166],[108,4626,65],{"className":4627},[171],". The sum always equals ",[108,4630,4632,4645],{"className":4631},[111],[108,4633,4635],{"className":4634},[115],[117,4636,4637],{"xmlns":119},[121,4638,4639,4643],{},[124,4640,4641],{},[127,4642,65],{},[151,4644,65],{"encoding":153},[108,4646,4648],{"className":4647,"ariaHidden":132},[158],[108,4649,4651,4654],{"className":4650},[162],[108,4652],{"className":4653,"style":237},[166],[108,4655,65],{"className":4656},[171],", because together all cases cover 100% of the possibilities.",[15,4659,4660],{},"The general formula we have just discovered works for any number of flips:",[15,4662,4664],{"className":4663},[394],[108,4665,4667,4717],{"className":4666},[111],[108,4668,4670],{"className":4669},[115],[117,4671,4672],{"xmlns":119},[121,4673,4674,4714],{},[124,4675,4676,4678,4680,4682,4685,4687,4690,4692,4694],{},[415,4677,3248],{},[130,4679,409],{"stretchy":853},[415,4681,420],{},[1886,4683,4684],{}," heads in ",[415,4686,417],{},[1886,4688,4689],{}," flips",[130,4691,423],{"stretchy":853},[130,4693,680],{},[411,4695,4696,4708],{},[124,4697,4698,4700,4706],{},[130,4699,409],{"fence":132},[411,4701,4702,4704],{"linethickness":413},[415,4703,417],{},[415,4705,420],{},[130,4707,423],{"fence":132},[2505,4709,4710,4712],{},[127,4711,675],{},[415,4713,417],{},[151,4715,4716],{"encoding":153},"P(k \\text{ heads in } n \\text{ flips}) = \\frac{\\binom{n}{k}}{2^n}",[108,4718,4720,4762],{"className":4719,"ariaHidden":132},[158],[108,4721,4723,4726,4729,4732,4735,4741,4744,4750,4753,4756,4759],{"className":4722},[162],[108,4724],{"className":4725,"style":3475},[166],[108,4727,3248],{"className":4728,"style":3275},[171,489],[108,4730,409],{"className":4731},[443],[108,4733,420],{"className":4734,"style":490},[171,489],[108,4736,4738],{"className":4737},[171,89],[108,4739,4684],{"className":4740},[171],[108,4742,417],{"className":4743},[171,489],[108,4745,4747],{"className":4746},[171,89],[108,4748,4689],{"className":4749},[171],[108,4751,423],{"className":4752},[523],[108,4754],{"className":4755,"style":766},[179],[108,4757,680],{"className":4758},[770],[108,4760],{"className":4761,"style":766},[179],[108,4763,4765,4768],{"className":4764},[162],[108,4766],{"className":4767,"style":3509},[166],[108,4769,4771,4774,4930],{"className":4770},[171],[108,4772],{"className":4773},[443,977],[108,4775,4777],{"className":4776},[411],[108,4778,4780,4922],{"className":4779},[457,458],[108,4781,4783,4919],{"className":4782},[462],[108,4784,4786,4827,4835],{"className":4785,"style":3528},[466],[108,4787,4788,4791],{"style":993},[108,4789],{"className":4790,"style":997},[474],[108,4792,4794],{"className":4793},[479,480,481,482],[108,4795,4797],{"className":4796},[171,482],[108,4798,4800,4803],{"className":4799},[171,482],[108,4801,675],{"className":4802},[171,482],[108,4804,4806],{"className":4805},[2533],[108,4807,4809],{"className":4808},[457],[108,4810,4812],{"className":4811},[462],[108,4813,4816],{"className":4814,"style":4815},[466],"height:0.5935em;",[108,4817,4818,4821],{"style":3561},[108,4819],{"className":4820,"style":3565},[474],[108,4822,4824],{"className":4823},[479,3569,450,482],[108,4825,417],{"className":4826},[171,489,482],[108,4828,4829,4832],{"style":1031},[108,4830],{"className":4831,"style":997},[474],[108,4833],{"className":4834,"style":1039},[1038],[108,4836,4837,4840],{"style":3583},[108,4838],{"className":4839,"style":997},[474],[108,4841,4843],{"className":4842},[479,480,481,482],[108,4844,4846],{"className":4845},[171,482],[108,4847,4849,4858,4910],{"className":4848},[171,482],[108,4850,4852],{"className":4851,"style":3600},[443,479,3569,3599,482,444],[108,4853,4855],{"className":4854},[449,450,482],[108,4856,409],{"className":4857},[482],[108,4859,4861],{"className":4860},[411],[108,4862,4864,4902],{"className":4863},[457,458],[108,4865,4867,4899],{"className":4866},[462],[108,4868,4871,4885],{"className":4869,"style":4870},[466],"height:0.7785em;",[108,4872,4873,4876],{"style":3622},[108,4874],{"className":4875,"style":3565},[474],[108,4877,4879],{"className":4878},[479,3569,450,482],[108,4880,4882],{"className":4881},[171,482],[108,4883,420],{"className":4884,"style":490},[171,489,482],[108,4886,4887,4890],{"style":3637},[108,4888],{"className":4889,"style":3565},[474],[108,4891,4893],{"className":4892},[479,3569,450,482],[108,4894,4896],{"className":4895},[171,482],[108,4897,417],{"className":4898},[171,489,482],[108,4900,510],{"className":4901},[509],[108,4903,4905],{"className":4904},[462],[108,4906,4908],{"className":4907,"style":3659},[466],[108,4909],{},[108,4911,4913],{"className":4912,"style":3600},[523,479,3569,3599,482,444],[108,4914,4916],{"className":4915},[449,450,482],[108,4917,423],{"className":4918},[482],[108,4920,510],{"className":4921},[509],[108,4923,4925],{"className":4924},[462],[108,4926,4928],{"className":4927,"style":517},[466],[108,4929],{},[108,4931],{"className":4932},[523,977],[19,4934,4936],{"id":4935},"case-study-5-flips","Case study: 5 flips",[15,4938,4939],{},"Let's raise the stakes. With 5 flips, the corresponding row of the triangle is:",[15,4941,4943],{"className":4942},[394],[108,4944,4946,4981],{"className":4945},[111],[108,4947,4949],{"className":4948},[115],[117,4950,4951],{"xmlns":119},[121,4952,4953,4978],{},[124,4954,4955,4957,4960,4962,4964,4966,4968,4970,4972,4974,4976],{},[127,4956,65],{},[179,4958],{"width":4959},"1em",[127,4961,2745],{},[179,4963],{"width":4959},[127,4965,2753],{},[179,4967],{"width":4959},[127,4969,2753],{},[179,4971],{"width":4959},[127,4973,2745],{},[179,4975],{"width":4959},[127,4977,65],{},[151,4979,4980],{"encoding":153},"1 \\quad 5 \\quad 10 \\quad 10 \\quad 5 \\quad 1",[108,4982,4984],{"className":4983,"ariaHidden":132},[158],[108,4985,4987,4990,4993,4997,5000,5003,5006,5009,5012,5015,5018,5021],{"className":4986},[162],[108,4988],{"className":4989,"style":237},[166],[108,4991,65],{"className":4992},[171],[108,4994],{"className":4995,"style":4996},[179],"margin-right:1em;",[108,4998,2745],{"className":4999},[171],[108,5001],{"className":5002,"style":4996},[179],[108,5004,2753],{"className":5005},[171],[108,5007],{"className":5008,"style":4996},[179],[108,5010,2753],{"className":5011},[171],[108,5013],{"className":5014,"style":4996},[179],[108,5016,2745],{"className":5017},[171],[108,5019],{"className":5020,"style":4996},[179],[108,5022,65],{"className":5023},[171],[15,5025,5026,5027,812],{},"and the total number of possible outcomes is ",[108,5028,5030,5052],{"className":5029},[111],[108,5031,5033],{"className":5032},[115],[117,5034,5035],{"xmlns":119},[121,5036,5037,5049],{},[124,5038,5039,5045,5047],{},[2505,5040,5041,5043],{},[127,5042,675],{},[127,5044,2745],{},[130,5046,680],{},[127,5048,2748],{},[151,5050,5051],{"encoding":153},"2^5 = 32",[108,5053,5055,5099],{"className":5054,"ariaHidden":132},[158],[108,5056,5058,5061,5090,5093,5096],{"className":5057},[162],[108,5059],{"className":5060,"style":2811},[166],[108,5062,5064,5067],{"className":5063},[171],[108,5065,675],{"className":5066},[171],[108,5068,5070],{"className":5069},[2533],[108,5071,5073],{"className":5072},[457],[108,5074,5076],{"className":5075},[462],[108,5077,5079],{"className":5078,"style":2811},[466],[108,5080,5081,5084],{"style":2545},[108,5082],{"className":5083,"style":475},[474],[108,5085,5087],{"className":5086},[479,480,481,482],[108,5088,2745],{"className":5089},[171,482],[108,5091],{"className":5092,"style":766},[179],[108,5094,680],{"className":5095},[770],[108,5097],{"className":5098,"style":766},[179],[108,5100,5102,5105],{"className":5101},[162],[108,5103],{"className":5104,"style":237},[166],[108,5106,2748],{"className":5107},[171],[15,5109,5110,5111,5114,5115,5239],{},"Let's look at some concrete examples. How many ways can we get exactly ",[27,5112,5113],{},"2 heads","? The triangle tells us ",[108,5116,5118,5146],{"className":5117},[111],[108,5119,5121],{"className":5120},[115],[117,5122,5123],{"xmlns":119},[121,5124,5125,5143],{},[124,5126,5127,5139,5141],{},[124,5128,5129,5131,5137],{},[130,5130,409],{"fence":132},[411,5132,5133,5135],{"linethickness":413},[127,5134,2745],{},[127,5136,675],{},[130,5138,423],{"fence":132},[130,5140,680],{},[127,5142,2753],{},[151,5144,5145],{"encoding":153},"\\binom{5}{2} = 10",[108,5147,5149,5230],{"className":5148,"ariaHidden":132},[158],[108,5150,5152,5155,5221,5224,5227],{"className":5151},[162],[108,5153],{"className":5154,"style":695},[166],[108,5156,5158,5164,5215],{"className":5157},[171],[108,5159,5161],{"className":5160,"style":445},[443,444],[108,5162,409],{"className":5163},[449,450],[108,5165,5167],{"className":5166},[411],[108,5168,5170,5207],{"className":5169},[457,458],[108,5171,5173,5204],{"className":5172},[462],[108,5174,5176,5190],{"className":5175,"style":717},[466],[108,5177,5178,5181],{"style":470},[108,5179],{"className":5180,"style":475},[474],[108,5182,5184],{"className":5183},[479,480,481,482],[108,5185,5187],{"className":5186},[171,482],[108,5188,675],{"className":5189},[171,482],[108,5191,5192,5195],{"style":493},[108,5193],{"className":5194,"style":475},[474],[108,5196,5198],{"className":5197},[479,480,481,482],[108,5199,5201],{"className":5200},[171,482],[108,5202,2745],{"className":5203},[171,482],[108,5205,510],{"className":5206},[509],[108,5208,5210],{"className":5209},[462],[108,5211,5213],{"className":5212,"style":517},[466],[108,5214],{},[108,5216,5218],{"className":5217,"style":445},[523,444],[108,5219,423],{"className":5220},[449,450],[108,5222],{"className":5223,"style":766},[179],[108,5225,680],{"className":5226},[770],[108,5228],{"className":5229,"style":766},[179],[108,5231,5233,5236],{"className":5232},[162],[108,5234],{"className":5235,"style":237},[166],[108,5237,2753],{"className":5238},[171],". Here are all 10 combinations:",[84,5241,5244],{"className":5242,"code":5243,"language":89,"meta":90},[87],"OOXXX   OXOXX   OXXOX   OXXXO   XOOXX\nXOXOX   XOXXO   XXOOX   XXOXO   XXXOO\n",[92,5245,5243],{"__ignoreMap":90},[15,5247,5248,5249,5252,5253,5377],{},"And exactly ",[27,5250,5251],{},"3 heads","? Also ",[108,5254,5256,5284],{"className":5255},[111],[108,5257,5259],{"className":5258},[115],[117,5260,5261],{"xmlns":119},[121,5262,5263,5281],{},[124,5264,5265,5277,5279],{},[124,5266,5267,5269,5275],{},[130,5268,409],{"fence":132},[411,5270,5271,5273],{"linethickness":413},[127,5272,2745],{},[127,5274,256],{},[130,5276,423],{"fence":132},[130,5278,680],{},[127,5280,2753],{},[151,5282,5283],{"encoding":153},"\\binom{5}{3} = 10",[108,5285,5287,5368],{"className":5286,"ariaHidden":132},[158],[108,5288,5290,5293,5359,5362,5365],{"className":5289},[162],[108,5291],{"className":5292,"style":695},[166],[108,5294,5296,5302,5353],{"className":5295},[171],[108,5297,5299],{"className":5298,"style":445},[443,444],[108,5300,409],{"className":5301},[449,450],[108,5303,5305],{"className":5304},[411],[108,5306,5308,5345],{"className":5307},[457,458],[108,5309,5311,5342],{"className":5310},[462],[108,5312,5314,5328],{"className":5313,"style":717},[466],[108,5315,5316,5319],{"style":470},[108,5317],{"className":5318,"style":475},[474],[108,5320,5322],{"className":5321},[479,480,481,482],[108,5323,5325],{"className":5324},[171,482],[108,5326,256],{"className":5327},[171,482],[108,5329,5330,5333],{"style":493},[108,5331],{"className":5332,"style":475},[474],[108,5334,5336],{"className":5335},[479,480,481,482],[108,5337,5339],{"className":5338},[171,482],[108,5340,2745],{"className":5341},[171,482],[108,5343,510],{"className":5344},[509],[108,5346,5348],{"className":5347},[462],[108,5349,5351],{"className":5350,"style":517},[466],[108,5352],{},[108,5354,5356],{"className":5355,"style":445},[523,444],[108,5357,423],{"className":5358},[449,450],[108,5360],{"className":5361,"style":766},[179],[108,5363,680],{"className":5364},[770],[108,5366],{"className":5367,"style":766},[179],[108,5369,5371,5374],{"className":5370},[162],[108,5372],{"className":5373,"style":237},[166],[108,5375,2753],{"className":5376},[171],". The triangle is symmetric: there are just as many ways to get 2 heads (and 3 tails) as 3 heads (and 2 tails).",[15,5379,5380],{},"The complete probability table:",[2617,5382,5383,5523],{},[2620,5384,5385],{},[2623,5386,5387,5417,5519,5521],{},[2626,5388,5389,3876],{"align":2628},[108,5390,5392,5405],{"className":5391},[111],[108,5393,5395],{"className":5394},[115],[117,5396,5397],{"xmlns":119},[121,5398,5399,5403],{},[124,5400,5401],{},[415,5402,420],{},[151,5404,420],{"encoding":153},[108,5406,5408],{"className":5407,"ariaHidden":132},[158],[108,5409,5411,5414],{"className":5410},[162],[108,5412],{"className":5413,"style":587},[166],[108,5415,420],{"className":5416,"style":490},[171,489],[2626,5418,5419],{"align":2628},[108,5420,5422,5444],{"className":5421},[111],[108,5423,5425],{"className":5424},[115],[117,5426,5427],{"xmlns":119},[121,5428,5429,5441],{},[124,5430,5431,5433,5439],{},[130,5432,409],{"fence":132},[411,5434,5435,5437],{"linethickness":413},[127,5436,2745],{},[415,5438,420],{},[130,5440,423],{"fence":132},[151,5442,5443],{"encoding":153},"\\binom{5}{k}",[108,5445,5447],{"className":5446,"ariaHidden":132},[158],[108,5448,5450,5453],{"className":5449},[162],[108,5451],{"className":5452,"style":695},[166],[108,5454,5456,5462,5513],{"className":5455},[171],[108,5457,5459],{"className":5458,"style":445},[443,444],[108,5460,409],{"className":5461},[449,450],[108,5463,5465],{"className":5464},[411],[108,5466,5468,5505],{"className":5467},[457,458],[108,5469,5471,5502],{"className":5470},[462],[108,5472,5474,5488],{"className":5473,"style":717},[466],[108,5475,5476,5479],{"style":470},[108,5477],{"className":5478,"style":475},[474],[108,5480,5482],{"className":5481},[479,480,481,482],[108,5483,5485],{"className":5484},[171,482],[108,5486,420],{"className":5487,"style":490},[171,489,482],[108,5489,5490,5493],{"style":493},[108,5491],{"className":5492,"style":475},[474],[108,5494,5496],{"className":5495},[479,480,481,482],[108,5497,5499],{"className":5498},[171,482],[108,5500,2745],{"className":5501},[171,482],[108,5503,510],{"className":5504},[509],[108,5506,5508],{"className":5507},[462],[108,5509,5511],{"className":5510,"style":517},[466],[108,5512],{},[108,5514,5516],{"className":5515,"style":445},[523,444],[108,5517,423],{"className":5518},[449,450],[2626,5520,3229],{"align":2628},[2626,5522,3983],{"align":2628},[2720,5524,5525,5634,5743,5852,5959,6066],{},[2623,5526,5527,5529,5531,5631],{},[2725,5528,1334],{"align":2628},[2725,5530,65],{"align":2628},[2725,5532,5533],{"align":2628},[108,5534,5536,5554],{"className":5535},[111],[108,5537,5539],{"className":5538},[115],[117,5540,5541],{"xmlns":119},[121,5542,5543,5551],{},[124,5544,5545],{},[411,5546,5547,5549],{},[127,5548,65],{},[127,5550,2748],{},[151,5552,5553],{"encoding":153},"\\frac{1}{32}",[108,5555,5557],{"className":5556,"ariaHidden":132},[158],[108,5558,5560,5563],{"className":5559},[162],[108,5561],{"className":5562,"style":1736},[166],[108,5564,5566,5569,5628],{"className":5565},[171],[108,5567],{"className":5568},[443,977],[108,5570,5572],{"className":5571},[411],[108,5573,5575,5620],{"className":5574},[457,458],[108,5576,5578,5617],{"className":5577},[462],[108,5579,5581,5595,5603],{"className":5580,"style":1663},[466],[108,5582,5583,5586],{"style":993},[108,5584],{"className":5585,"style":997},[474],[108,5587,5589],{"className":5588},[479,480,481,482],[108,5590,5592],{"className":5591},[171,482],[108,5593,2748],{"className":5594},[171,482],[108,5596,5597,5600],{"style":1031},[108,5598],{"className":5599,"style":997},[474],[108,5601],{"className":5602,"style":1039},[1038],[108,5604,5605,5608],{"style":1042},[108,5606],{"className":5607,"style":997},[474],[108,5609,5611],{"className":5610},[479,480,481,482],[108,5612,5614],{"className":5613},[171,482],[108,5615,65],{"className":5616},[171,482],[108,5618,510],{"className":5619},[509],[108,5621,5623],{"className":5622},[462],[108,5624,5626],{"className":5625,"style":517},[466],[108,5627],{},[108,5629],{"className":5630},[523,977],[2725,5632,5633],{"align":2628},"3.1%",[2623,5635,5636,5638,5640,5740],{},[2725,5637,65],{"align":2628},[2725,5639,2745],{"align":2628},[2725,5641,5642],{"align":2628},[108,5643,5645,5663],{"className":5644},[111],[108,5646,5648],{"className":5647},[115],[117,5649,5650],{"xmlns":119},[121,5651,5652,5660],{},[124,5653,5654],{},[411,5655,5656,5658],{},[127,5657,2745],{},[127,5659,2748],{},[151,5661,5662],{"encoding":153},"\\frac{5}{32}",[108,5664,5666],{"className":5665,"ariaHidden":132},[158],[108,5667,5669,5672],{"className":5668},[162],[108,5670],{"className":5671,"style":1736},[166],[108,5673,5675,5678,5737],{"className":5674},[171],[108,5676],{"className":5677},[443,977],[108,5679,5681],{"className":5680},[411],[108,5682,5684,5729],{"className":5683},[457,458],[108,5685,5687,5726],{"className":5686},[462],[108,5688,5690,5704,5712],{"className":5689,"style":1663},[466],[108,5691,5692,5695],{"style":993},[108,5693],{"className":5694,"style":997},[474],[108,5696,5698],{"className":5697},[479,480,481,482],[108,5699,5701],{"className":5700},[171,482],[108,5702,2748],{"className":5703},[171,482],[108,5705,5706,5709],{"style":1031},[108,5707],{"className":5708,"style":997},[474],[108,5710],{"className":5711,"style":1039},[1038],[108,5713,5714,5717],{"style":1042},[108,5715],{"className":5716,"style":997},[474],[108,5718,5720],{"className":5719},[479,480,481,482],[108,5721,5723],{"className":5722},[171,482],[108,5724,2745],{"className":5725},[171,482],[108,5727,510],{"className":5728},[509],[108,5730,5732],{"className":5731},[462],[108,5733,5735],{"className":5734,"style":517},[466],[108,5736],{},[108,5738],{"className":5739},[523,977],[2725,5741,5742],{"align":2628},"15.6%",[2623,5744,5745,5747,5749,5849],{},[2725,5746,675],{"align":2628},[2725,5748,2753],{"align":2628},[2725,5750,5751],{"align":2628},[108,5752,5754,5772],{"className":5753},[111],[108,5755,5757],{"className":5756},[115],[117,5758,5759],{"xmlns":119},[121,5760,5761,5769],{},[124,5762,5763],{},[411,5764,5765,5767],{},[127,5766,2753],{},[127,5768,2748],{},[151,5770,5771],{"encoding":153},"\\frac{10}{32}",[108,5773,5775],{"className":5774,"ariaHidden":132},[158],[108,5776,5778,5781],{"className":5777},[162],[108,5779],{"className":5780,"style":1736},[166],[108,5782,5784,5787,5846],{"className":5783},[171],[108,5785],{"className":5786},[443,977],[108,5788,5790],{"className":5789},[411],[108,5791,5793,5838],{"className":5792},[457,458],[108,5794,5796,5835],{"className":5795},[462],[108,5797,5799,5813,5821],{"className":5798,"style":1663},[466],[108,5800,5801,5804],{"style":993},[108,5802],{"className":5803,"style":997},[474],[108,5805,5807],{"className":5806},[479,480,481,482],[108,5808,5810],{"className":5809},[171,482],[108,5811,2748],{"className":5812},[171,482],[108,5814,5815,5818],{"style":1031},[108,5816],{"className":5817,"style":997},[474],[108,5819],{"className":5820,"style":1039},[1038],[108,5822,5823,5826],{"style":1042},[108,5824],{"className":5825,"style":997},[474],[108,5827,5829],{"className":5828},[479,480,481,482],[108,5830,5832],{"className":5831},[171,482],[108,5833,2753],{"className":5834},[171,482],[108,5836,510],{"className":5837},[509],[108,5839,5841],{"className":5840},[462],[108,5842,5844],{"className":5843,"style":517},[466],[108,5845],{},[108,5847],{"className":5848},[523,977],[2725,5850,5851],{"align":2628},"31.3%",[2623,5853,5854,5856,5858,5957],{},[2725,5855,256],{"align":2628},[2725,5857,2753],{"align":2628},[2725,5859,5860],{"align":2628},[108,5861,5863,5880],{"className":5862},[111],[108,5864,5866],{"className":5865},[115],[117,5867,5868],{"xmlns":119},[121,5869,5870,5878],{},[124,5871,5872],{},[411,5873,5874,5876],{},[127,5875,2753],{},[127,5877,2748],{},[151,5879,5771],{"encoding":153},[108,5881,5883],{"className":5882,"ariaHidden":132},[158],[108,5884,5886,5889],{"className":5885},[162],[108,5887],{"className":5888,"style":1736},[166],[108,5890,5892,5895,5954],{"className":5891},[171],[108,5893],{"className":5894},[443,977],[108,5896,5898],{"className":5897},[411],[108,5899,5901,5946],{"className":5900},[457,458],[108,5902,5904,5943],{"className":5903},[462],[108,5905,5907,5921,5929],{"className":5906,"style":1663},[466],[108,5908,5909,5912],{"style":993},[108,5910],{"className":5911,"style":997},[474],[108,5913,5915],{"className":5914},[479,480,481,482],[108,5916,5918],{"className":5917},[171,482],[108,5919,2748],{"className":5920},[171,482],[108,5922,5923,5926],{"style":1031},[108,5924],{"className":5925,"style":997},[474],[108,5927],{"className":5928,"style":1039},[1038],[108,5930,5931,5934],{"style":1042},[108,5932],{"className":5933,"style":997},[474],[108,5935,5937],{"className":5936},[479,480,481,482],[108,5938,5940],{"className":5939},[171,482],[108,5941,2753],{"className":5942},[171,482],[108,5944,510],{"className":5945},[509],[108,5947,5949],{"className":5948},[462],[108,5950,5952],{"className":5951,"style":517},[466],[108,5953],{},[108,5955],{"className":5956},[523,977],[2725,5958,5851],{"align":2628},[2623,5960,5961,5963,5965,6064],{},[2725,5962,136],{"align":2628},[2725,5964,2745],{"align":2628},[2725,5966,5967],{"align":2628},[108,5968,5970,5987],{"className":5969},[111],[108,5971,5973],{"className":5972},[115],[117,5974,5975],{"xmlns":119},[121,5976,5977,5985],{},[124,5978,5979],{},[411,5980,5981,5983],{},[127,5982,2745],{},[127,5984,2748],{},[151,5986,5662],{"encoding":153},[108,5988,5990],{"className":5989,"ariaHidden":132},[158],[108,5991,5993,5996],{"className":5992},[162],[108,5994],{"className":5995,"style":1736},[166],[108,5997,5999,6002,6061],{"className":5998},[171],[108,6000],{"className":6001},[443,977],[108,6003,6005],{"className":6004},[411],[108,6006,6008,6053],{"className":6007},[457,458],[108,6009,6011,6050],{"className":6010},[462],[108,6012,6014,6028,6036],{"className":6013,"style":1663},[466],[108,6015,6016,6019],{"style":993},[108,6017],{"className":6018,"style":997},[474],[108,6020,6022],{"className":6021},[479,480,481,482],[108,6023,6025],{"className":6024},[171,482],[108,6026,2748],{"className":6027},[171,482],[108,6029,6030,6033],{"style":1031},[108,6031],{"className":6032,"style":997},[474],[108,6034],{"className":6035,"style":1039},[1038],[108,6037,6038,6041],{"style":1042},[108,6039],{"className":6040,"style":997},[474],[108,6042,6044],{"className":6043},[479,480,481,482],[108,6045,6047],{"className":6046},[171,482],[108,6048,2745],{"className":6049},[171,482],[108,6051,510],{"className":6052},[509],[108,6054,6056],{"className":6055},[462],[108,6057,6059],{"className":6058,"style":517},[466],[108,6060],{},[108,6062],{"className":6063},[523,977],[2725,6065,5742],{"align":2628},[2623,6067,6068,6070,6072,6171],{},[2725,6069,2745],{"align":2628},[2725,6071,65],{"align":2628},[2725,6073,6074],{"align":2628},[108,6075,6077,6094],{"className":6076},[111],[108,6078,6080],{"className":6079},[115],[117,6081,6082],{"xmlns":119},[121,6083,6084,6092],{},[124,6085,6086],{},[411,6087,6088,6090],{},[127,6089,65],{},[127,6091,2748],{},[151,6093,5553],{"encoding":153},[108,6095,6097],{"className":6096,"ariaHidden":132},[158],[108,6098,6100,6103],{"className":6099},[162],[108,6101],{"className":6102,"style":1736},[166],[108,6104,6106,6109,6168],{"className":6105},[171],[108,6107],{"className":6108},[443,977],[108,6110,6112],{"className":6111},[411],[108,6113,6115,6160],{"className":6114},[457,458],[108,6116,6118,6157],{"className":6117},[462],[108,6119,6121,6135,6143],{"className":6120,"style":1663},[466],[108,6122,6123,6126],{"style":993},[108,6124],{"className":6125,"style":997},[474],[108,6127,6129],{"className":6128},[479,480,481,482],[108,6130,6132],{"className":6131},[171,482],[108,6133,2748],{"className":6134},[171,482],[108,6136,6137,6140],{"style":1031},[108,6138],{"className":6139,"style":997},[474],[108,6141],{"className":6142,"style":1039},[1038],[108,6144,6145,6148],{"style":1042},[108,6146],{"className":6147,"style":997},[474],[108,6149,6151],{"className":6150},[479,480,481,482],[108,6152,6154],{"className":6153},[171,482],[108,6155,65],{"className":6156},[171,482],[108,6158,510],{"className":6159},[509],[108,6161,6163],{"className":6162},[462],[108,6164,6166],{"className":6165,"style":517},[466],[108,6167],{},[108,6169],{"className":6170},[523,977],[2725,6172,5633],{"align":2628},[15,6174,6175],{},"If we sketch this as a histogram, the distribution has a familiar shape:",[84,6177,6180],{"className":6178,"code":6179,"language":89,"meta":90},[87],"0 heads  ▌              3.1%\n1 head   ▌▌▌▌▌         15.6%\n2 heads  ▌▌▌▌▌▌▌▌▌▌    31.3%\n3 heads  ▌▌▌▌▌▌▌▌▌▌    31.3%\n4 heads  ▌▌▌▌▌         15.6%\n5 heads  ▌              3.1%\n",[92,6181,6179],{"__ignoreMap":90},[15,6183,6184],{},"Notice three things:",[57,6186,6187,6193,6199],{},[60,6188,6189,6192],{},[27,6190,6191],{},"Symmetry."," The distribution is a perfect mirror around the center. Getting 0 heads is just as unlikely as getting 5; getting 1 is just as likely as getting 4.",[60,6194,6195,6198],{},[27,6196,6197],{},"Central concentration."," The most probable results cluster around the center (2 or 3 heads), while the extremes (all heads or all tails) are very rare.",[60,6200,6201,6204],{},[27,6202,6203],{},"Bell shape."," The silhouette resembles the famous Gaussian curve. This is no accident: as we increase the number of flips, the distribution increasingly approximates that curve.",[15,6206,6207,6208,812],{},"If you are interested in exploring more of the statistics behind coin flips, read ",[2764,6209,6212],{"href":6210,"rel":6211},"https:\u002F\u002Fflipthecoin.app\u002Fblog\u002Fthe-unexpected-statistics-of-coin-flips\u002F",[2768],"The Unexpected Statistics of Coin Flips",[19,6214,6216],{"id":6215},"the-algebraic-connection-the-binomial-theorem","The algebraic connection: the binomial theorem",[15,6218,6219,6220,6223],{},"So far we have talked about triangles and coins. But there is a third protagonist that ties everything together: ",[27,6221,6222],{},"algebra",". Specifically, an expression you may remember from school:",[15,6225,6227],{"className":6226},[394],[108,6228,6230,6257],{"className":6229},[111],[108,6231,6233],{"className":6232},[115],[117,6234,6235],{"xmlns":119},[121,6236,6237,6254],{},[124,6238,6239,6241,6243,6245,6248],{},[130,6240,409],{"stretchy":853},[415,6242,2764],{},[130,6244,259],{},[415,6246,6247],{},"b",[2505,6249,6250,6252],{},[130,6251,423],{"stretchy":853},[415,6253,417],{},[151,6255,6256],{"encoding":153},"(a + b)^n",[108,6258,6260,6281],{"className":6259,"ariaHidden":132},[158],[108,6261,6263,6266,6269,6272,6275,6278],{"className":6262},[162],[108,6264],{"className":6265,"style":3475},[166],[108,6267,409],{"className":6268},[443],[108,6270,2764],{"className":6271},[171,489],[108,6273],{"className":6274,"style":281},[179],[108,6276,259],{"className":6277},[285],[108,6279],{"className":6280,"style":281},[179],[108,6282,6284,6287,6290],{"className":6283},[162],[108,6285],{"className":6286,"style":3475},[166],[108,6288,6247],{"className":6289},[171,489],[108,6291,6293,6296],{"className":6292},[523],[108,6294,423],{"className":6295},[523],[108,6297,6299],{"className":6298},[2533],[108,6300,6302],{"className":6301},[457],[108,6303,6305],{"className":6304},[462],[108,6306,6308],{"className":6307,"style":2523},[466],[108,6309,6310,6313],{"style":2545},[108,6311],{"className":6312,"style":475},[474],[108,6314,6316],{"className":6315},[479,480,481,482],[108,6317,417],{"className":6318},[171,489,482],[15,6320,6321],{},"Before expanding it, let's clarify a couple of concepts:",[6323,6324,6325,6447],"ul",{},[60,6326,6327,6328,6331,6332,812],{},"A ",[27,6329,6330],{},"polynomial"," is a mathematical expression made up of sums of terms with variables raised to various powers. For example: ",[108,6333,6335,6366],{"className":6334},[111],[108,6336,6338],{"className":6337},[115],[117,6339,6340],{"xmlns":119},[121,6341,6342,6363],{},[124,6343,6344,6346,6353,6355,6357,6359,6361],{},[127,6345,256],{},[2505,6347,6348,6351],{},[415,6349,6350],{},"x",[127,6352,675],{},[130,6354,259],{},[127,6356,675],{},[415,6358,6350],{},[130,6360,259],{},[127,6362,65],{},[151,6364,6365],{"encoding":153},"3x^2 + 2x + 1",[108,6367,6369,6417,6438],{"className":6368,"ariaHidden":132},[158],[108,6370,6372,6376,6379,6408,6411,6414],{"className":6371},[162],[108,6373],{"className":6374,"style":6375},[166],"height:0.8974em;vertical-align:-0.0833em;",[108,6377,256],{"className":6378},[171],[108,6380,6382,6385],{"className":6381},[171],[108,6383,6350],{"className":6384},[171,489],[108,6386,6388],{"className":6387},[2533],[108,6389,6391],{"className":6390},[457],[108,6392,6394],{"className":6393},[462],[108,6395,6397],{"className":6396,"style":2811},[466],[108,6398,6399,6402],{"style":2545},[108,6400],{"className":6401,"style":475},[474],[108,6403,6405],{"className":6404},[479,480,481,482],[108,6406,675],{"className":6407},[171,482],[108,6409],{"className":6410,"style":281},[179],[108,6412,259],{"className":6413},[285],[108,6415],{"className":6416,"style":281},[179],[108,6418,6420,6423,6426,6429,6432,6435],{"className":6419},[162],[108,6421],{"className":6422,"style":274},[166],[108,6424,675],{"className":6425},[171],[108,6427,6350],{"className":6428},[171,489],[108,6430],{"className":6431,"style":281},[179],[108,6433,259],{"className":6434},[285],[108,6436],{"className":6437,"style":281},[179],[108,6439,6441,6444],{"className":6440},[162],[108,6442],{"className":6443,"style":237},[166],[108,6445,65],{"className":6446},[171],[60,6448,6327,6449,6452,6453,6456,6457,812],{},[27,6450,6451],{},"binomial"," is a special case of a polynomial: it has exactly ",[27,6454,6455],{},"two"," terms. For example: ",[108,6458,6460,6482],{"className":6459},[111],[108,6461,6463],{"className":6462},[115],[117,6464,6465],{"xmlns":119},[121,6466,6467,6479],{},[124,6468,6469,6471,6473,6475,6477],{},[130,6470,409],{"stretchy":853},[415,6472,2764],{},[130,6474,259],{},[415,6476,6247],{},[130,6478,423],{"stretchy":853},[151,6480,6481],{"encoding":153},"(a + b)",[108,6483,6485,6506],{"className":6484,"ariaHidden":132},[158],[108,6486,6488,6491,6494,6497,6500,6503],{"className":6487},[162],[108,6489],{"className":6490,"style":3475},[166],[108,6492,409],{"className":6493},[443],[108,6495,2764],{"className":6496},[171,489],[108,6498],{"className":6499,"style":281},[179],[108,6501,259],{"className":6502},[285],[108,6504],{"className":6505,"style":281},[179],[108,6507,6509,6512,6515],{"className":6508},[162],[108,6510],{"className":6511,"style":3475},[166],[108,6513,6247],{"className":6514},[171,489],[108,6516,423],{"className":6517},[523],[15,6519,6520,6521,6549],{},"Raising a binomial to a power means multiplying it by itself ",[108,6522,6524,6537],{"className":6523},[111],[108,6525,6527],{"className":6526},[115],[117,6528,6529],{"xmlns":119},[121,6530,6531,6535],{},[124,6532,6533],{},[415,6534,417],{},[151,6536,417],{"encoding":153},[108,6538,6540],{"className":6539,"ariaHidden":132},[158],[108,6541,6543,6546],{"className":6542},[162],[108,6544],{"className":6545,"style":557},[166],[108,6547,417],{"className":6548},[171,489]," times. Let's start with the simplest cases:",[15,6551,6552],{},[27,6553,6554],{},"Power 2:",[15,6556,6558],{"className":6557},[394],[108,6559,6561,6611],{"className":6560},[111],[108,6562,6564],{"className":6563},[115],[117,6565,6566],{"xmlns":119},[121,6567,6568,6608],{},[124,6569,6570,6572,6574,6576,6578,6584,6586,6592,6594,6596,6598,6600,6602],{},[130,6571,409],{"stretchy":853},[415,6573,2764],{},[130,6575,259],{},[415,6577,6247],{},[2505,6579,6580,6582],{},[130,6581,423],{"stretchy":853},[127,6583,675],{},[130,6585,680],{},[2505,6587,6588,6590],{},[415,6589,2764],{},[127,6591,675],{},[130,6593,259],{},[127,6595,675],{},[415,6597,2764],{},[415,6599,6247],{},[130,6601,259],{},[2505,6603,6604,6606],{},[415,6605,6247],{},[127,6607,675],{},[151,6609,6610],{"encoding":153},"(a + b)^2 = a^2 + 2ab + b^2",[108,6612,6614,6635,6683,6727,6750],{"className":6613,"ariaHidden":132},[158],[108,6615,6617,6620,6623,6626,6629,6632],{"className":6616},[162],[108,6618],{"className":6619,"style":3475},[166],[108,6621,409],{"className":6622},[443],[108,6624,2764],{"className":6625},[171,489],[108,6627],{"className":6628,"style":281},[179],[108,6630,259],{"className":6631},[285],[108,6633],{"className":6634,"style":281},[179],[108,6636,6638,6642,6645,6674,6677,6680],{"className":6637},[162],[108,6639],{"className":6640,"style":6641},[166],"height:1.0641em;vertical-align:-0.25em;",[108,6643,6247],{"className":6644},[171,489],[108,6646,6648,6651],{"className":6647},[523],[108,6649,423],{"className":6650},[523],[108,6652,6654],{"className":6653},[2533],[108,6655,6657],{"className":6656},[457],[108,6658,6660],{"className":6659},[462],[108,6661,6663],{"className":6662,"style":2811},[466],[108,6664,6665,6668],{"style":2545},[108,6666],{"className":6667,"style":475},[474],[108,6669,6671],{"className":6670},[479,480,481,482],[108,6672,675],{"className":6673},[171,482],[108,6675],{"className":6676,"style":766},[179],[108,6678,680],{"className":6679},[770],[108,6681],{"className":6682,"style":766},[179],[108,6684,6686,6689,6718,6721,6724],{"className":6685},[162],[108,6687],{"className":6688,"style":6375},[166],[108,6690,6692,6695],{"className":6691},[171],[108,6693,2764],{"className":6694},[171,489],[108,6696,6698],{"className":6697},[2533],[108,6699,6701],{"className":6700},[457],[108,6702,6704],{"className":6703},[462],[108,6705,6707],{"className":6706,"style":2811},[466],[108,6708,6709,6712],{"style":2545},[108,6710],{"className":6711,"style":475},[474],[108,6713,6715],{"className":6714},[479,480,481,482],[108,6716,675],{"className":6717},[171,482],[108,6719],{"className":6720,"style":281},[179],[108,6722,259],{"className":6723},[285],[108,6725],{"className":6726,"style":281},[179],[108,6728,6730,6734,6737,6741,6744,6747],{"className":6729},[162],[108,6731],{"className":6732,"style":6733},[166],"height:0.7778em;vertical-align:-0.0833em;",[108,6735,675],{"className":6736},[171],[108,6738,6740],{"className":6739},[171,489],"ab",[108,6742],{"className":6743,"style":281},[179],[108,6745,259],{"className":6746},[285],[108,6748],{"className":6749,"style":281},[179],[108,6751,6753,6756],{"className":6752},[162],[108,6754],{"className":6755,"style":2811},[166],[108,6757,6759,6762],{"className":6758},[171],[108,6760,6247],{"className":6761},[171,489],[108,6763,6765],{"className":6764},[2533],[108,6766,6768],{"className":6767},[457],[108,6769,6771],{"className":6770},[462],[108,6772,6774],{"className":6773,"style":2811},[466],[108,6775,6776,6779],{"style":2545},[108,6777],{"className":6778,"style":475},[474],[108,6780,6782],{"className":6781},[479,480,481,482],[108,6783,675],{"className":6784},[171,482],[15,6786,6787,6788,6843],{},"The coefficients are ",[108,6789,6791,6813],{"className":6790},[111],[108,6792,6794],{"className":6793},[115],[117,6795,6796],{"xmlns":119},[121,6797,6798,6810],{},[124,6799,6800,6802,6804,6806,6808],{},[127,6801,65],{},[130,6803,133],{"separator":132},[127,6805,675],{},[130,6807,133],{"separator":132},[127,6809,65],{},[151,6811,6812],{"encoding":153},"1, 2, 1",[108,6814,6816],{"className":6815,"ariaHidden":132},[158],[108,6817,6819,6822,6825,6828,6831,6834,6837,6840],{"className":6818},[162],[108,6820],{"className":6821,"style":167},[166],[108,6823,65],{"className":6824},[171],[108,6826,133],{"className":6827},[175],[108,6829],{"className":6830,"style":180},[179],[108,6832,675],{"className":6833},[171],[108,6835,133],{"className":6836},[175],[108,6838],{"className":6839,"style":180},[179],[108,6841,65],{"className":6842},[171]," — row 2 of the triangle.",[15,6845,6846],{},[27,6847,6848],{},"Power 3:",[15,6850,6852],{"className":6851},[394],[108,6853,6855,6921],{"className":6854},[111],[108,6856,6858],{"className":6857},[115],[117,6859,6860],{"xmlns":119},[121,6861,6862,6918],{},[124,6863,6864,6866,6868,6870,6872,6878,6880,6886,6888,6890,6896,6898,6900,6902,6904,6910,6912],{},[130,6865,409],{"stretchy":853},[415,6867,2764],{},[130,6869,259],{},[415,6871,6247],{},[2505,6873,6874,6876],{},[130,6875,423],{"stretchy":853},[127,6877,256],{},[130,6879,680],{},[2505,6881,6882,6884],{},[415,6883,2764],{},[127,6885,256],{},[130,6887,259],{},[127,6889,256],{},[2505,6891,6892,6894],{},[415,6893,2764],{},[127,6895,675],{},[415,6897,6247],{},[130,6899,259],{},[127,6901,256],{},[415,6903,2764],{},[2505,6905,6906,6908],{},[415,6907,6247],{},[127,6909,675],{},[130,6911,259],{},[2505,6913,6914,6916],{},[415,6915,6247],{},[127,6917,256],{},[151,6919,6920],{"encoding":153},"(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3",[108,6922,6924,6945,6992,7036,7086,7136],{"className":6923,"ariaHidden":132},[158],[108,6925,6927,6930,6933,6936,6939,6942],{"className":6926},[162],[108,6928],{"className":6929,"style":3475},[166],[108,6931,409],{"className":6932},[443],[108,6934,2764],{"className":6935},[171,489],[108,6937],{"className":6938,"style":281},[179],[108,6940,259],{"className":6941},[285],[108,6943],{"className":6944,"style":281},[179],[108,6946,6948,6951,6954,6983,6986,6989],{"className":6947},[162],[108,6949],{"className":6950,"style":6641},[166],[108,6952,6247],{"className":6953},[171,489],[108,6955,6957,6960],{"className":6956},[523],[108,6958,423],{"className":6959},[523],[108,6961,6963],{"className":6962},[2533],[108,6964,6966],{"className":6965},[457],[108,6967,6969],{"className":6968},[462],[108,6970,6972],{"className":6971,"style":2811},[466],[108,6973,6974,6977],{"style":2545},[108,6975],{"className":6976,"style":475},[474],[108,6978,6980],{"className":6979},[479,480,481,482],[108,6981,256],{"className":6982},[171,482],[108,6984],{"className":6985,"style":766},[179],[108,6987,680],{"className":6988},[770],[108,6990],{"className":6991,"style":766},[179],[108,6993,6995,6998,7027,7030,7033],{"className":6994},[162],[108,6996],{"className":6997,"style":6375},[166],[108,6999,7001,7004],{"className":7000},[171],[108,7002,2764],{"className":7003},[171,489],[108,7005,7007],{"className":7006},[2533],[108,7008,7010],{"className":7009},[457],[108,7011,7013],{"className":7012},[462],[108,7014,7016],{"className":7015,"style":2811},[466],[108,7017,7018,7021],{"style":2545},[108,7019],{"className":7020,"style":475},[474],[108,7022,7024],{"className":7023},[479,480,481,482],[108,7025,256],{"className":7026},[171,482],[108,7028],{"className":7029,"style":281},[179],[108,7031,259],{"className":7032},[285],[108,7034],{"className":7035,"style":281},[179],[108,7037,7039,7042,7045,7074,7077,7080,7083],{"className":7038},[162],[108,7040],{"className":7041,"style":6375},[166],[108,7043,256],{"className":7044},[171],[108,7046,7048,7051],{"className":7047},[171],[108,7049,2764],{"className":7050},[171,489],[108,7052,7054],{"className":7053},[2533],[108,7055,7057],{"className":7056},[457],[108,7058,7060],{"className":7059},[462],[108,7061,7063],{"className":7062,"style":2811},[466],[108,7064,7065,7068],{"style":2545},[108,7066],{"className":7067,"style":475},[474],[108,7069,7071],{"className":7070},[479,480,481,482],[108,7072,675],{"className":7073},[171,482],[108,7075,6247],{"className":7076},[171,489],[108,7078],{"className":7079,"style":281},[179],[108,7081,259],{"className":7082},[285],[108,7084],{"className":7085,"style":281},[179],[108,7087,7089,7092,7095,7098,7127,7130,7133],{"className":7088},[162],[108,7090],{"className":7091,"style":6375},[166],[108,7093,256],{"className":7094},[171],[108,7096,2764],{"className":7097},[171,489],[108,7099,7101,7104],{"className":7100},[171],[108,7102,6247],{"className":7103},[171,489],[108,7105,7107],{"className":7106},[2533],[108,7108,7110],{"className":7109},[457],[108,7111,7113],{"className":7112},[462],[108,7114,7116],{"className":7115,"style":2811},[466],[108,7117,7118,7121],{"style":2545},[108,7119],{"className":7120,"style":475},[474],[108,7122,7124],{"className":7123},[479,480,481,482],[108,7125,675],{"className":7126},[171,482],[108,7128],{"className":7129,"style":281},[179],[108,7131,259],{"className":7132},[285],[108,7134],{"className":7135,"style":281},[179],[108,7137,7139,7142],{"className":7138},[162],[108,7140],{"className":7141,"style":2811},[166],[108,7143,7145,7148],{"className":7144},[171],[108,7146,6247],{"className":7147},[171,489],[108,7149,7151],{"className":7150},[2533],[108,7152,7154],{"className":7153},[457],[108,7155,7157],{"className":7156},[462],[108,7158,7160],{"className":7159,"style":2811},[466],[108,7161,7162,7165],{"style":2545},[108,7163],{"className":7164,"style":475},[474],[108,7166,7168],{"className":7167},[479,480,481,482],[108,7169,256],{"className":7170},[171,482],[15,7172,6787,7173,7240],{},[108,7174,7176,7201],{"className":7175},[111],[108,7177,7179],{"className":7178},[115],[117,7180,7181],{"xmlns":119},[121,7182,7183,7199],{},[124,7184,7185,7187,7189,7191,7193,7195,7197],{},[127,7186,65],{},[130,7188,133],{"separator":132},[127,7190,256],{},[130,7192,133],{"separator":132},[127,7194,256],{},[130,7196,133],{"separator":132},[127,7198,65],{},[151,7200,2957],{"encoding":153},[108,7202,7204],{"className":7203,"ariaHidden":132},[158],[108,7205,7207,7210,7213,7216,7219,7222,7225,7228,7231,7234,7237],{"className":7206},[162],[108,7208],{"className":7209,"style":167},[166],[108,7211,65],{"className":7212},[171],[108,7214,133],{"className":7215},[175],[108,7217],{"className":7218,"style":180},[179],[108,7220,256],{"className":7221},[171],[108,7223,133],{"className":7224},[175],[108,7226],{"className":7227,"style":180},[179],[108,7229,256],{"className":7230},[171],[108,7232,133],{"className":7233},[175],[108,7235],{"className":7236,"style":180},[179],[108,7238,65],{"className":7239},[171]," — row 3.",[15,7242,7243],{},[27,7244,7245],{},"Power 5:",[15,7247,7249],{"className":7248},[394],[108,7250,7252,7350],{"className":7251},[111],[108,7253,7255],{"className":7254},[115],[117,7256,7257],{"xmlns":119},[121,7258,7259,7347],{},[124,7260,7261,7263,7265,7267,7269,7275,7277,7283,7285,7287,7293,7295,7297,7299,7305,7311,7313,7315,7321,7327,7329,7331,7333,7339,7341],{},[130,7262,409],{"stretchy":853},[415,7264,2764],{},[130,7266,259],{},[415,7268,6247],{},[2505,7270,7271,7273],{},[130,7272,423],{"stretchy":853},[127,7274,2745],{},[130,7276,680],{},[2505,7278,7279,7281],{},[415,7280,2764],{},[127,7282,2745],{},[130,7284,259],{},[127,7286,2745],{},[2505,7288,7289,7291],{},[415,7290,2764],{},[127,7292,136],{},[415,7294,6247],{},[130,7296,259],{},[127,7298,2753],{},[2505,7300,7301,7303],{},[415,7302,2764],{},[127,7304,256],{},[2505,7306,7307,7309],{},[415,7308,6247],{},[127,7310,675],{},[130,7312,259],{},[127,7314,2753],{},[2505,7316,7317,7319],{},[415,7318,2764],{},[127,7320,675],{},[2505,7322,7323,7325],{},[415,7324,6247],{},[127,7326,256],{},[130,7328,259],{},[127,7330,2745],{},[415,7332,2764],{},[2505,7334,7335,7337],{},[415,7336,6247],{},[127,7338,136],{},[130,7340,259],{},[2505,7342,7343,7345],{},[415,7344,6247],{},[127,7346,2745],{},[151,7348,7349],{"encoding":153},"(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + 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= \\sum_{k=0}^{n} \\binom{n}{k} \\, a^{n-k} \\, 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0",[108,8290,8292,8310],{"className":8291,"ariaHidden":132},[158],[108,8293,8295,8298,8301,8304,8307],{"className":8294},[162],[108,8296],{"className":8297,"style":587},[166],[108,8299,420],{"className":8300,"style":490},[171,489],[108,8302],{"className":8303,"style":766},[179],[108,8305,680],{"className":8306},[770],[108,8308],{"className":8309,"style":766},[179],[108,8311,8313,8316],{"className":8312},[162],[108,8314],{"className":8315,"style":237},[166],[108,8317,1334],{"className":8318},[171]," to ",[108,8321,8323,8341],{"className":8322},[111],[108,8324,8326],{"className":8325},[115],[117,8327,8328],{"xmlns":119},[121,8329,8330,8338],{},[124,8331,8332,8334,8336],{},[415,8333,420],{},[130,8335,680],{},[415,8337,417],{},[151,8339,8340],{"encoding":153},"k = n",[108,8342,8344,8362],{"className":8343,"ariaHidden":132},[158],[108,8345,8347,8350,8353,8356,8359],{"className":8346},[162],[108,8348],{"className":8349,"style":587},[166],[108,8351,420],{"className":8352,"style":490},[171,489],[108,8354],{"className":8355,"style":766},[179],[108,8357,680],{"className":8358},[770],[108,8360],{"className":8361,"style":766},[179],[108,8363,8365,8368],{"className":8364},[162],[108,8366],{"className":8367,"style":557},[166],[108,8369,417],{"className":8370},[171,489],"». Each term has a binomial coefficient ",[108,8373,8375,8396],{"className":8374},[111],[108,8376,8378],{"className":8377},[115],[117,8379,8380],{"xmlns":119},[121,8381,8382,8394],{},[124,8383,8384,8386,8392],{},[130,8385,409],{"fence":132},[411,8387,8388,8390],{"linethickness":413},[415,8389,417],{},[415,8391,420],{},[130,8393,423],{"fence":132},[151,8395,426],{"encoding":153},[108,8397,8399],{"className":8398,"ariaHidden":132},[158],[108,8400,8402,8405],{"className":8401},[162],[108,8403],{"className":8404,"style":436},[166],[108,8406,8408,8414,8465],{"className":8407},[171],[108,8409,8411],{"className":8410,"style":445},[443,444],[108,8412,409],{"className":8413},[449,450],[108,8415,8417],{"className":8416},[411],[108,8418,8420,8457],{"className":8419},[457,458],[108,8421,8423,8454],{"className":8422},[462],[108,8424,8426,8440],{"className":8425,"style":467},[466],[108,8427,8428,8431],{"style":470},[108,8429],{"className":8430,"style":475},[474],[108,8432,8434],{"className":8433},[479,480,481,482],[108,8435,8437],{"className":8436},[171,482],[108,8438,420],{"className":8439,"style":490},[171,489,482],[108,8441,8442,8445],{"style":493},[108,8443],{"className":8444,"style":475},[474],[108,8446,8448],{"className":8447},[479,480,481,482],[108,8449,8451],{"className":8450},[171,482],[108,8452,417],{"className":8453},[171,489,482],[108,8455,510],{"className":8456},[509],[108,8458,8460],{"className":8459},[462],[108,8461,8463],{"className":8462,"style":517},[466],[108,8464],{},[108,8466,8468],{"className":8467,"style":445},[523,444],[108,8469,423],{"className":8470},[449,450]," that comes directly from the triangle.",[380,8473,8475],{"id":8474},"the-bridge-between-algebra-and-probability","The bridge between algebra and probability",[15,8477,8478,8479,8507,8508,8510,8511,8507,8539,2488],{},"Now comes the fascinating part. Substitute ",[108,8480,8482,8495],{"className":8481},[111],[108,8483,8485],{"className":8484},[115],[117,8486,8487],{"xmlns":119},[121,8488,8489,8493],{},[124,8490,8491],{},[415,8492,2764],{},[151,8494,2764],{"encoding":153},[108,8496,8498],{"className":8497,"ariaHidden":132},[158],[108,8499,8501,8504],{"className":8500},[162],[108,8502],{"className":8503,"style":557},[166],[108,8505,2764],{"className":8506},[171,489]," with the probability of getting ",[27,8509,2277],{}," and ",[108,8512,8514,8527],{"className":8513},[111],[108,8515,8517],{"className":8516},[115],[117,8518,8519],{"xmlns":119},[121,8520,8521,8525],{},[124,8522,8523],{},[415,8524,6247],{},[151,8526,6247],{"encoding":153},[108,8528,8530],{"className":8529,"ariaHidden":132},[158],[108,8531,8533,8536],{"className":8532},[162],[108,8534],{"className":8535,"style":587},[166],[108,8537,6247],{"className":8538},[171,489],[27,8540,2281],{},[15,8542,8544],{"className":8543},[394],[108,8545,8547,8603],{"className":8546},[111],[108,8548,8550],{"className":8549},[115],[117,8551,8552],{"xmlns":119},[121,8553,8554,8600],{},[124,8555,8556,8558,8560,8562,8564,8566,8568,8570,8576,8578,8580,8582,8584,8586,8588,8590,8592,8594],{},[415,8557,2764],{},[130,8559,680],{},[415,8561,3248],{},[130,8563,409],{"stretchy":853},[1886,8565,2277],{},[130,8567,423],{"stretchy":853},[130,8569,680],{},[411,8571,8572,8574],{},[127,8573,65],{},[127,8575,675],{},[130,8577,133],{"separator":132},[179,8579],{"width":4959},[415,8581,6247],{},[130,8583,680],{},[415,8585,3248],{},[130,8587,409],{"stretchy":853},[1886,8589,2281],{},[130,8591,423],{"stretchy":853},[130,8593,680],{},[411,8595,8596,8598],{},[127,8597,65],{},[127,8599,675],{},[151,8601,8602],{"encoding":153},"a = P(\\text{heads}) = \\frac{1}{2}, \\quad b = P(\\text{tails}) = \\frac{1}{2}",[108,8604,8606,8624,8654,8749,8779],{"className":8605,"ariaHidden":132},[158],[108,8607,8609,8612,8615,8618,8621],{"className":8608},[162],[108,8610],{"className":8611,"style":557},[166],[108,8613,2764],{"className":8614},[171,489],[108,8616],{"className":8617,"style":766},[179],[108,8619,680],{"className":8620},[770],[108,8622],{"className":8623,"style":766},[179],[108,8625,8627,8630,8633,8636,8642,8645,8648,8651],{"className":8626},[162],[108,8628],{"className":8629,"style":3475},[166],[108,8631,3248],{"className":8632,"style":3275},[171,489],[108,8634,409],{"className":8635},[443],[108,8637,8639],{"className":8638},[171,89],[108,8640,2277],{"className":8641},[171],[108,8643,423],{"className":8644},[523],[108,8646],{"className":8647,"style":766},[179],[108,8649,680],{"className":8650},[770],[108,8652],{"className":8653,"style":766},[179],[108,8655,8657,8660,8728,8731,8734,8737,8740,8743,8746],{"className":8656},[162],[108,8658],{"className":8659,"style":1736},[166],[108,8661,8663,8666,8725],{"className":8662},[171],[108,8664],{"className":8665},[443,977],[108,8667,8669],{"className":8668},[411],[108,8670,8672,8717],{"className":8671},[457,458],[108,8673,8675,8714],{"className":8674},[462],[108,8676,8678,8692,8700],{"className":8677,"style":1663},[466],[108,8679,8680,8683],{"style":993},[108,8681],{"className":8682,"style":997},[474],[108,8684,8686],{"className":8685},[479,480,481,482],[108,8687,8689],{"className":8688},[171,482],[108,8690,675],{"className":8691},[171,482],[108,8693,8694,8697],{"style":1031},[108,8695],{"className":8696,"style":997},[474],[108,8698],{"className":8699,"style":1039},[1038],[108,8701,8702,8705],{"style":1042},[108,8703],{"className":8704,"style":997},[474],[108,8706,8708],{"className":8707},[479,480,481,482],[108,8709,8711],{"className":8710},[171,482],[108,8712,65],{"className":8713},[171,482],[108,8715,510],{"className":8716},[509],[108,8718,8720],{"className":8719},[462],[108,8721,8723],{"className":8722,"style":517},[466],[108,8724],{},[108,8726],{"className":8727},[523,977],[108,8729,133],{"className":8730},[175],[108,8732],{"className":8733,"style":4996},[179],[108,8735],{"className":8736,"style":180},[179],[108,8738,6247],{"className":8739},[171,489],[108,8741],{"className":8742,"style":766},[179],[108,8744,680],{"className":8745},[770],[108,8747],{"className":8748,"style":766},[179],[108,8750,8752,8755,8758,8761,8767,8770,8773,8776],{"className":8751},[162],[108,8753],{"className":8754,"style":3475},[166],[108,8756,3248],{"className":8757,"style":3275},[171,489],[108,8759,409],{"className":8760},[443],[108,8762,8764],{"className":8763},[171,89],[108,8765,2281],{"className":8766},[171],[108,8768,423],{"className":8769},[523],[108,8771],{"className":8772,"style":766},[179],[108,8774,680],{"className":8775},[770],[108,8777],{"className":8778,"style":766},[179],[108,8780,8782,8785],{"className":8781},[162],[108,8783],{"className":8784,"style":1736},[166],[108,8786,8788,8791,8850],{"className":8787},[171],[108,8789],{"className":8790},[443,977],[108,8792,8794],{"className":8793},[411],[108,8795,8797,8842],{"className":8796},[457,458],[108,8798,8800,8839],{"className":8799},[462],[108,8801,8803,8817,8825],{"className":8802,"style":1663},[466],[108,8804,8805,8808],{"style":993},[108,8806],{"className":8807,"style":997},[474],[108,8809,8811],{"className":8810},[479,480,481,482],[108,8812,8814],{"className":8813},[171,482],[108,8815,675],{"className":8816},[171,482],[108,8818,8819,8822],{"style":1031},[108,8820],{"className":8821,"style":997},[474],[108,8823],{"className":8824,"style":1039},[1038],[108,8826,8827,8830],{"style":1042},[108,8828],{"className":8829,"style":997},[474],[108,8831,8833],{"className":8832},[479,480,481,482],[108,8834,8836],{"className":8835},[171,482],[108,8837,65],{"className":8838},[171,482],[108,8840,510],{"className":8841},[509],[108,8843,8845],{"className":8844},[462],[108,8846,8848],{"className":8847,"style":517},[466],[108,8849],{},[108,8851],{"className":8852},[523,977],[15,8854,8855],{},"The binomial becomes:",[15,8857,8859],{"className":8858},[394],[108,8860,8862,9000],{"className":8861},[111],[108,8863,8865],{"className":8864},[115],[117,8866,8867],{"xmlns":119},[121,8868,8869,8997],{},[124,8870,8871,8895,8897,8911,8923,8945,8961,8963,8977],{},[2505,8872,8873,8893],{},[124,8874,8875,8877,8883,8885,8891],{},[130,8876,409],{"fence":132},[411,8878,8879,8881],{},[127,8880,65],{},[127,8882,675],{},[130,8884,259],{},[411,8886,8887,8889],{},[127,8888,65],{},[127,8890,675],{},[130,8892,423],{"fence":132},[415,8894,417],{},[130,8896,680],{},[7891,8898,8899,8901,8909],{},[130,8900,7895],{},[124,8902,8903,8905,8907],{},[415,8904,420],{},[130,8906,680],{},[127,8908,1334],{},[415,8910,417],{},[124,8912,8913,8915,8921],{},[130,8914,409],{"fence":132},[411,8916,8917,8919],{"linethickness":413},[415,8918,417],{},[415,8920,420],{},[130,8922,423],{"fence":132},[2505,8924,8925,8937],{},[124,8926,8927,8929,8935],{},[130,8928,409],{"fence":132},[411,8930,8931,8933],{},[127,8932,65],{},[127,8934,675],{},[130,8936,423],{"fence":132},[124,8938,8939,8941,8943],{},[415,8940,417],{},[130,8942,870],{},[415,8944,420],{},[2505,8946,8947,8959],{},[124,8948,8949,8951,8957],{},[130,8950,409],{"fence":132},[411,8952,8953,8955],{},[127,8954,65],{},[127,8956,675],{},[130,8958,423],{"fence":132},[415,8960,420],{},[130,8962,680],{},[7891,8964,8965,8967,8975],{},[130,8966,7895],{},[124,8968,8969,8971,8973],{},[415,8970,420],{},[130,8972,680],{},[127,8974,1334],{},[415,8976,417],{},[411,8978,8979,8991],{},[124,8980,8981,8983,8989],{},[130,8982,409],{"fence":132},[411,8984,8985,8987],{"linethickness":413},[415,8986,417],{},[415,8988,420],{},[130,8990,423],{"fence":132},[2505,8992,8993,8995],{},[127,8994,675],{},[415,8996,417],{},[151,8998,8999],{"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}",[108,9001,9003,9207,9589],{"className":9002,"ariaHidden":132},[158],[108,9004,9006,9010,9198,9201,9204],{"className":9005},[162],[108,9007],{"className":9008,"style":9009},[166],"height:1.2543em;vertical-align:-0.35em;",[108,9011,9013,9173],{"className":9012},[2180],[108,9014,9016,9022,9090,9093,9096,9099,9167],{"className":9015},[2180],[108,9017,9019],{"className":9018,"style":445},[443,444],[108,9020,409],{"className":9021},[449,450],[108,9023,9025,9028,9087],{"className":9024},[171],[108,9026],{"className":9027},[443,977],[108,9029,9031],{"className":9030},[411],[108,9032,9034,9079],{"className":9033},[457,458],[108,9035,9037,9076],{"className":9036},[462],[108,9038,9040,9054,9062],{"className":9039,"style":1663},[466],[108,9041,9042,9045],{"style":993},[108,9043],{"className":9044,"style":997},[474],[108,9046,9048],{"className":9047},[479,480,481,482],[108,9049,9051],{"className":9050},[171,482],[108,9052,675],{"className":9053},[171,482],[108,9055,9056,9059],{"style":1031},[108,9057],{"className":9058,"style":997},[474],[108,9060],{"className":9061,"style":1039},[1038],[108,9063,9064,9067],{"style":1042},[108,9065],{"className":9066,"style":997},[474],[108,9068,9070],{"className":9069},[479,480,481,482],[108,9071,9073],{"className":9072},[171,482],[108,9074,65],{"className":9075},[171,482],[108,9077,510],{"className":9078},[509],[108,9080,9082],{"className":9081},[462],[108,9083,9085],{"className":9084,"style":517},[466],[108,9086],{},[108,9088],{"className":9089},[523,977],[108,9091],{"className":9092,"style":281},[179],[108,9094,259],{"className":9095},[285],[108,9097],{"className":9098,"style":281},[179],[108,9100,9102,9105,9164],{"className":9101},[171],[108,9103],{"className":9104},[443,977],[108,9106,9108],{"className":9107},[411],[108,9109,9111,9156],{"className":9110},[457,458],[108,9112,9114,9153],{"className":9113},[462],[108,9115,9117,9131,9139],{"className":9116,"style":1663},[466],[108,9118,9119,9122],{"style":993},[108,9120],{"className":9121,"style":997},[474],[108,9123,9125],{"className":9124},[479,480,481,482],[108,9126,9128],{"className":9127},[171,482],[108,9129,675],{"className":9130},[171,482],[108,9132,9133,9136],{"style":1031},[108,9134],{"className":9135,"style":997},[474],[108,9137],{"className":9138,"style":1039},[1038],[108,9140,9141,9144],{"style":1042},[108,9142],{"className":9143,"style":997},[474],[108,9145,9147],{"className":9146},[479,480,481,482],[108,9148,9150],{"className":9149},[171,482],[108,9151,65],{"className":9152},[171,482],[108,9154,510],{"className":9155},[509],[108,9157,9159],{"className":9158},[462],[108,9160,9162],{"className":9161,"style":517},[466],[108,9163],{},[108,9165],{"className":9166},[523,977],[108,9168,9170],{"className":9169,"style":445},[523,444],[108,9171,423],{"className":9172},[449,450],[108,9174,9176],{"className":9175},[2533],[108,9177,9179],{"className":9178},[457],[108,9180,9182],{"className":9181},[462],[108,9183,9186],{"className":9184,"style":9185},[466],"height:0.9043em;",[108,9187,9189,9192],{"style":9188},"top:-3.3029em;margin-right:0.05em;",[108,9190],{"className":9191,"style":475},[474],[108,9193,9195],{"className":9194},[479,480,481,482],[108,9196,417],{"className":9197},[171,489,482],[108,9199],{"className":9200,"style":766},[179],[108,9202,680],{"className":9203},[770],[108,9205],{"className":9206,"style":766},[179],[108,9208,9210,9214,9277,9280,9346,9349,9468,9471,9580,9583,9586],{"className":9209},[162],[108,9211],{"className":9212,"style":9213},[166],"height:1.439em;vertical-align:-0.35em;",[108,9215,9217,9220],{"className":9216},[8024],[108,9218,7895],{"className":9219,"style":8030},[8024,8028,8029],[108,9221,9223],{"className":9222},[2533],[108,9224,9226,9269],{"className":9225},[457,458],[108,9227,9229,9266],{"className":9228},[462],[108,9230,9232,9252],{"className":9231,"style":8043},[466],[108,9233,9234,9237],{"style":8046},[108,9235],{"className":9236,"style":475},[474],[108,9238,9240],{"className":9239},[479,480,481,482],[108,9241,9243,9246,9249],{"className":9242},[171,482],[108,9244,420],{"className":9245,"style":490},[171,489,482],[108,9247,680],{"className":9248},[770,482],[108,9250,1334],{"className":9251},[171,482],[108,9253,9254,9257],{"style":8067},[108,9255],{"className":9256,"style":475},[474],[108,9258,9260],{"className":9259},[479,480,481,482],[108,9261,9263],{"className":9262},[171,482],[108,9264,417],{"className":9265},[171,489,482],[108,9267,510],{"className":9268},[509],[108,9270,9272],{"className":9271},[462],[108,9273,9275],{"className":9274,"style":8089},[466],[108,9276],{},[108,9278],{"className":9279,"style":180},[179],[108,9281,9283,9289,9340],{"className":9282},[171],[108,9284,9286],{"className":9285,"style":445},[443,444],[108,9287,409],{"className":9288},[449,450],[108,9290,9292],{"className":9291},[411],[108,9293,9295,9332],{"className":9294},[457,458],[108,9296,9298,9329],{"className":9297},[462],[108,9299,9301,9315],{"className":9300,"style":467},[466],[108,9302,9303,9306],{"style":470},[108,9304],{"className":9305,"style":475},[474],[108,9307,9309],{"className":9308},[479,480,481,482],[108,9310,9312],{"className":9311},[171,482],[108,9313,420],{"className":9314,"style":490},[171,489,482],[108,9316,9317,9320],{"style":493},[108,9318],{"className":9319,"style":475},[474],[108,9321,9323],{"className":9322},[479,480,481,482],[108,9324,9326],{"className":9325},[171,482],[108,9327,417],{"className":9328},[171,489,482],[108,9330,510],{"className":9331},[509],[108,9333,9335],{"className":9334},[462],[108,9336,9338],{"className":9337,"style":517},[466],[108,9339],{},[108,9341,9343],{"className":9342,"style":445},[523,444],[108,9344,423],{"className":9345},[449,450],[108,9347],{"className":9348,"style":180},[179],[108,9350,9352,9435],{"className":9351},[2180],[108,9353,9355,9361,9429],{"className":9354},[2180],[108,9356,9358],{"className":9357,"style":445},[443,444],[108,9359,409],{"className":9360},[449,450],[108,9362,9364,9367,9426],{"className":9363},[171],[108,9365],{"className":9366},[443,977],[108,9368,9370],{"className":9369},[411],[108,9371,9373,9418],{"className":9372},[457,458],[108,9374,9376,9415],{"className":9375},[462],[108,9377,9379,9393,9401],{"className":9378,"style":1663},[466],[108,9380,9381,9384],{"style":993},[108,9382],{"className":9383,"style":997},[474],[108,9385,9387],{"className":9386},[479,480,481,482],[108,9388,9390],{"className":9389},[171,482],[108,9391,675],{"className":9392},[171,482],[108,9394,9395,9398],{"style":1031},[108,9396],{"className":9397,"style":997},[474],[108,9399],{"className":9400,"style":1039},[1038],[108,9402,9403,9406],{"style":1042},[108,9404],{"className":9405,"style":997},[474],[108,9407,9409],{"className":9408},[479,480,481,482],[108,9410,9412],{"className":9411},[171,482],[108,9413,65],{"className":9414},[171,482],[108,9416,510],{"className":9417},[509],[108,9419,9421],{"className":9420},[462],[108,9422,9424],{"className":9423,"style":517},[466],[108,9425],{},[108,9427],{"className":9428},[523,977],[108,9430,9432],{"className":9431,"style":445},[523,444],[108,9433,423],{"className":9434},[449,450],[108,9436,9438],{"className":9437},[2533],[108,9439,9441],{"className":9440},[457],[108,9442,9444],{"className":9443},[462],[108,9445,9448],{"className":9446,"style":9447},[466],"height:1.089em;",[108,9449,9450,9453],{"style":9188},[108,9451],{"className":9452,"style":475},[474],[108,9454,9456],{"className":9455},[479,480,481,482],[108,9457,9459,9462,9465],{"className":9458},[171,482],[108,9460,417],{"className":9461},[171,489,482],[108,9463,870],{"className":9464},[285,482],[108,9466,420],{"className":9467,"style":490},[171,489,482],[108,9469],{"className":9470,"style":180},[179],[108,9472,9474,9557],{"className":9473},[2180],[108,9475,9477,9483,9551],{"className":9476},[2180],[108,9478,9480],{"className":9479,"style":445},[443,444],[108,9481,409],{"className":9482},[449,450],[108,9484,9486,9489,9548],{"className":9485},[171],[108,9487],{"className":9488},[443,977],[108,9490,9492],{"className":9491},[411],[108,9493,9495,9540],{"className":9494},[457,458],[108,9496,9498,9537],{"className":9497},[462],[108,9499,9501,9515,9523],{"className":9500,"style":1663},[466],[108,9502,9503,9506],{"style":993},[108,9504],{"className":9505,"style":997},[474],[108,9507,9509],{"className":9508},[479,480,481,482],[108,9510,9512],{"className":9511},[171,482],[108,9513,675],{"className":9514},[171,482],[108,9516,9517,9520],{"style":1031},[108,9518],{"className":9519,"style":997},[474],[108,9521],{"className":9522,"style":1039},[1038],[108,9524,9525,9528],{"style":1042},[108,9526],{"className":9527,"style":997},[474],[108,9529,9531],{"className":9530},[479,480,481,482],[108,9532,9534],{"className":9533},[171,482],[108,9535,65],{"className":9536},[171,482],[108,9538,510],{"className":9539},[509],[108,9541,9543],{"className":9542},[462],[108,9544,9546],{"className":9545,"style":517},[466],[108,9547],{},[108,9549],{"className":9550},[523,977],[108,9552,9554],{"className":9553,"style":445},[523,444],[108,9555,423],{"className":9556},[449,450],[108,9558,9560],{"className":9559},[2533],[108,9561,9563],{"className":9562},[457],[108,9564,9566],{"className":9565},[462],[108,9567,9569],{"className":9568,"style":9447},[466],[108,9570,9571,9574],{"style":9188},[108,9572],{"className":9573,"style":475},[474],[108,9575,9577],{"className":9576},[479,480,481,482],[108,9578,420],{"className":9579,"style":490},[171,489,482],[108,9581],{"className":9582,"style":766},[179],[108,9584,680],{"className":9585},[770],[108,9587],{"className":9588,"style":766},[179],[108,9590,9592,9595,9658,9661],{"className":9591},[162],[108,9593],{"className":9594,"style":3509},[166],[108,9596,9598,9601],{"className":9597},[8024],[108,9599,7895],{"className":9600,"style":8030},[8024,8028,8029],[108,9602,9604],{"className":9603},[2533],[108,9605,9607,9650],{"className":9606},[457,458],[108,9608,9610,9647],{"className":9609},[462],[108,9611,9613,9633],{"className":9612,"style":8043},[466],[108,9614,9615,9618],{"style":8046},[108,9616],{"className":9617,"style":475},[474],[108,9619,9621],{"className":9620},[479,480,481,482],[108,9622,9624,9627,9630],{"className":9623},[171,482],[108,9625,420],{"className":9626,"style":490},[171,489,482],[108,9628,680],{"className":9629},[770,482],[108,9631,1334],{"className":9632},[171,482],[108,9634,9635,9638],{"style":8067},[108,9636],{"className":9637,"style":475},[474],[108,9639,9641],{"className":9640},[479,480,481,482],[108,9642,9644],{"className":9643},[171,482],[108,9645,417],{"className":9646},[171,489,482],[108,9648,510],{"className":9649},[509],[108,9651,9653],{"className":9652},[462],[108,9654,9656],{"className":9655,"style":8089},[466],[108,9657],{},[108,9659],{"className":9660,"style":180},[179],[108,9662,9664,9667,9821],{"className":9663},[171],[108,9665],{"className":9666},[443,977],[108,9668,9670],{"className":9669},[411],[108,9671,9673,9813],{"className":9672},[457,458],[108,9674,9676,9810],{"className":9675},[462],[108,9677,9679,9719,9727],{"className":9678,"style":3528},[466],[108,9680,9681,9684],{"style":993},[108,9682],{"className":9683,"style":997},[474],[108,9685,9687],{"className":9686},[479,480,481,482],[108,9688,9690],{"className":9689},[171,482],[108,9691,9693,9696],{"className":9692},[171,482],[108,9694,675],{"className":9695},[171,482],[108,9697,9699],{"className":9698},[2533],[108,9700,9702],{"className":9701},[457],[108,9703,9705],{"className":9704},[462],[108,9706,9708],{"className":9707,"style":4815},[466],[108,9709,9710,9713],{"style":3561},[108,9711],{"className":9712,"style":3565},[474],[108,9714,9716],{"className":9715},[479,3569,450,482],[108,9717,417],{"className":9718},[171,489,482],[108,9720,9721,9724],{"style":1031},[108,9722],{"className":9723,"style":997},[474],[108,9725],{"className":9726,"style":1039},[1038],[108,9728,9729,9732],{"style":3583},[108,9730],{"className":9731,"style":997},[474],[108,9733,9735],{"className":9734},[479,480,481,482],[108,9736,9738],{"className":9737},[171,482],[108,9739,9741,9750,9801],{"className":9740},[171,482],[108,9742,9744],{"className":9743,"style":3600},[443,479,3569,3599,482,444],[108,9745,9747],{"className":9746},[449,450,482],[108,9748,409],{"className":9749},[482],[108,9751,9753],{"className":9752},[411],[108,9754,9756,9793],{"className":9755},[457,458],[108,9757,9759,9790],{"className":9758},[462],[108,9760,9762,9776],{"className":9761,"style":4870},[466],[108,9763,9764,9767],{"style":3622},[108,9765],{"className":9766,"style":3565},[474],[108,9768,9770],{"className":9769},[479,3569,450,482],[108,9771,9773],{"className":9772},[171,482],[108,9774,420],{"className":9775,"style":490},[171,489,482],[108,9777,9778,9781],{"style":3637},[108,9779],{"className":9780,"style":3565},[474],[108,9782,9784],{"className":9783},[479,3569,450,482],[108,9785,9787],{"className":9786},[171,482],[108,9788,417],{"className":9789},[171,489,482],[108,9791,510],{"className":9792},[509],[108,9794,9796],{"className":9795},[462],[108,9797,9799],{"className":9798,"style":3659},[466],[108,9800],{},[108,9802,9804],{"className":9803,"style":3600},[523,479,3569,3599,482,444],[108,9805,9807],{"className":9806},[449,450,482],[108,9808,423],{"className":9809},[482],[108,9811,510],{"className":9812},[509],[108,9814,9816],{"className":9815},[462],[108,9817,9819],{"className":9818,"style":517},[466],[108,9820],{},[108,9822],{"className":9823},[523,977],[15,9825,9826,9827,9908,9909,10116,10117,10145,10146,10198],{},"The left side is simply ",[108,9828,9830,9852],{"className":9829},[111],[108,9831,9833],{"className":9832},[115],[117,9834,9835],{"xmlns":119},[121,9836,9837,9849],{},[124,9838,9839,9845,9847],{},[2505,9840,9841,9843],{},[127,9842,65],{},[415,9844,417],{},[130,9846,680],{},[127,9848,65],{},[151,9850,9851],{"encoding":153},"1^n = 1",[108,9853,9855,9899],{"className":9854,"ariaHidden":132},[158],[108,9856,9858,9861,9890,9893,9896],{"className":9857},[162],[108,9859],{"className":9860,"style":2523},[166],[108,9862,9864,9867],{"className":9863},[171],[108,9865,65],{"className":9866},[171],[108,9868,9870],{"className":9869},[2533],[108,9871,9873],{"className":9872},[457],[108,9874,9876],{"className":9875},[462],[108,9877,9879],{"className":9878,"style":2523},[466],[108,9880,9881,9884],{"style":2545},[108,9882],{"className":9883,"style":475},[474],[108,9885,9887],{"className":9886},[479,480,481,482],[108,9888,417],{"className":9889},[171,489,482],[108,9891],{"className":9892,"style":766},[179],[108,9894,680],{"className":9895},[770],[108,9897],{"className":9898,"style":766},[179],[108,9900,9902,9905],{"className":9901},[162],[108,9903],{"className":9904,"style":237},[166],[108,9906,65],{"className":9907},[171],": total certainty. The right side is the sum of all individual probabilities. And each term ",[108,9910,9912,9944],{"className":9911},[111],[108,9913,9915],{"className":9914},[115],[117,9916,9917],{"xmlns":119},[121,9918,9919,9941],{},[124,9920,9921],{},[411,9922,9923,9935],{},[124,9924,9925,9927,9933],{},[130,9926,409],{"fence":132},[411,9928,9929,9931],{"linethickness":413},[415,9930,417],{},[415,9932,420],{},[130,9934,423],{"fence":132},[2505,9936,9937,9939],{},[127,9938,675],{},[415,9940,417],{},[151,9942,9943],{"encoding":153},"\\frac{\\binom{n}{k}}{2^n}",[108,9945,9947],{"className":9946,"ariaHidden":132},[158],[108,9948,9950,9953],{"className":9949},[162],[108,9951],{"className":9952,"style":3509},[166],[108,9954,9956,9959,10113],{"className":9955},[171],[108,9957],{"className":9958},[443,977],[108,9960,9962],{"className":9961},[411],[108,9963,9965,10105],{"className":9964},[457,458],[108,9966,9968,10102],{"className":9967},[462],[108,9969,9971,10011,10019],{"className":9970,"style":3528},[466],[108,9972,9973,9976],{"style":993},[108,9974],{"className":9975,"style":997},[474],[108,9977,9979],{"className":9978},[479,480,481,482],[108,9980,9982],{"className":9981},[171,482],[108,9983,9985,9988],{"className":9984},[171,482],[108,9986,675],{"className":9987},[171,482],[108,9989,9991],{"className":9990},[2533],[108,9992,9994],{"className":9993},[457],[108,9995,9997],{"className":9996},[462],[108,9998,10000],{"className":9999,"style":4815},[466],[108,10001,10002,10005],{"style":3561},[108,10003],{"className":10004,"style":3565},[474],[108,10006,10008],{"className":10007},[479,3569,450,482],[108,10009,417],{"className":10010},[171,489,482],[108,10012,10013,10016],{"style":1031},[108,10014],{"className":10015,"style":997},[474],[108,10017],{"className":10018,"style":1039},[1038],[108,10020,10021,10024],{"style":3583},[108,10022],{"className":10023,"style":997},[474],[108,10025,10027],{"className":10026},[479,480,481,482],[108,10028,10030],{"className":10029},[171,482],[108,10031,10033,10042,10093],{"className":10032},[171,482],[108,10034,10036],{"className":10035,"style":3600},[443,479,3569,3599,482,444],[108,10037,10039],{"className":10038},[449,450,482],[108,10040,409],{"className":10041},[482],[108,10043,10045],{"className":10044},[411],[108,10046,10048,10085],{"className":10047},[457,458],[108,10049,10051,10082],{"className":10050},[462],[108,10052,10054,10068],{"className":10053,"style":4870},[466],[108,10055,10056,10059],{"style":3622},[108,10057],{"className":10058,"style":3565},[474],[108,10060,10062],{"className":10061},[479,3569,450,482],[108,10063,10065],{"className":10064},[171,482],[108,10066,420],{"className":10067,"style":490},[171,489,482],[108,10069,10070,10073],{"style":3637},[108,10071],{"className":10072,"style":3565},[474],[108,10074,10076],{"className":10075},[479,3569,450,482],[108,10077,10079],{"className":10078},[171,482],[108,10080,417],{"className":10081},[171,489,482],[108,10083,510],{"className":10084},[509],[108,10086,10088],{"className":10087},[462],[108,10089,10091],{"className":10090,"style":3659},[466],[108,10092],{},[108,10094,10096],{"className":10095,"style":3600},[523,479,3569,3599,482,444],[108,10097,10099],{"className":10098},[449,450,482],[108,10100,423],{"className":10101},[482],[108,10103,510],{"className":10104},[509],[108,10106,10108],{"className":10107},[462],[108,10109,10111],{"className":10110,"style":517},[466],[108,10112],{},[108,10114],{"className":10115},[523,977]," is exactly the probability of getting ",[108,10118,10120,10133],{"className":10119},[111],[108,10121,10123],{"className":10122},[115],[117,10124,10125],{"xmlns":119},[121,10126,10127,10131],{},[124,10128,10129],{},[415,10130,420],{},[151,10132,420],{"encoding":153},[108,10134,10136],{"className":10135,"ariaHidden":132},[158],[108,10137,10139,10142],{"className":10138},[162],[108,10140],{"className":10141,"style":587},[166],[108,10143,420],{"className":10144,"style":490},[171,489]," tails (or equivalently, ",[108,10147,10149,10167],{"className":10148},[111],[108,10150,10152],{"className":10151},[115],[117,10153,10154],{"xmlns":119},[121,10155,10156,10164],{},[124,10157,10158,10160,10162],{},[415,10159,417],{},[130,10161,870],{},[415,10163,420],{},[151,10165,10166],{"encoding":153},"n - k",[108,10168,10170,10189],{"className":10169,"ariaHidden":132},[158],[108,10171,10173,10177,10180,10183,10186],{"className":10172},[162],[108,10174],{"className":10175,"style":10176},[166],"height:0.6667em;vertical-align:-0.0833em;",[108,10178,417],{"className":10179},[171,489],[108,10181],{"className":10182,"style":281},[179],[108,10184,870],{"className":10185},[285],[108,10187],{"className":10188,"style":281},[179],[108,10190,10192,10195],{"className":10191},[162],[108,10193],{"className":10194,"style":587},[166],[108,10196,420],{"className":10197,"style":490},[171,489]," heads).",[15,10200,10201],{},"Let's verify with 5 flips:",[15,10203,10205],{"className":10204},[394],[108,10206,10208,10260],{"className":10207},[111],[108,10209,10211],{"className":10210},[115],[117,10212,10213],{"xmlns":119},[121,10214,10215,10257],{},[124,10216,10217,10245,10247,10253,10255],{},[411,10218,10219,10243],{},[124,10220,10221,10223,10225,10227,10229,10231,10233,10235,10237,10239,10241],{},[127,10222,65],{},[130,10224,259],{},[127,10226,2745],{},[130,10228,259],{},[127,10230,2753],{},[130,10232,259],{},[127,10234,2753],{},[130,10236,259],{},[127,10238,2745],{},[130,10240,259],{},[127,10242,65],{},[127,10244,2748],{},[130,10246,680],{},[411,10248,10249,10251],{},[127,10250,2748],{},[127,10252,2748],{},[130,10254,680],{},[127,10256,65],{},[151,10258,10259],{"encoding":153},"\\frac{1 + 5 + 10 + 10 + 5 + 1}{32} = \\frac{32}{32} = 1",[108,10261,10263,10376,10459],{"className":10262,"ariaHidden":132},[158],[108,10264,10266,10269,10367,10370,10373],{"className":10265},[162],[108,10267],{"className":10268,"style":1736},[166],[108,10270,10272,10275,10364],{"className":10271},[171],[108,10273],{"className":10274},[443,977],[108,10276,10278],{"className":10277},[411],[108,10279,10281,10356],{"className":10280},[457,458],[108,10282,10284,10353],{"className":10283},[462],[108,10285,10287,10301,10309],{"className":10286,"style":1663},[466],[108,10288,10289,10292],{"style":993},[108,10290],{"className":10291,"style":997},[474],[108,10293,10295],{"className":10294},[479,480,481,482],[108,10296,10298],{"className":10297},[171,482],[108,10299,2748],{"className":10300},[171,482],[108,10302,10303,10306],{"style":1031},[108,10304],{"className":10305,"style":997},[474],[108,10307],{"className":10308,"style":1039},[1038],[108,10310,10311,10314],{"style":1042},[108,10312],{"className":10313,"style":997},[474],[108,10315,10317],{"className":10316},[479,480,481,482],[108,10318,10320,10323,10326,10329,10332,10335,10338,10341,10344,10347,10350],{"className":10319},[171,482],[108,10321,65],{"className":10322},[171,482],[108,10324,259],{"className":10325},[285,482],[108,10327,2745],{"className":10328},[171,482],[108,10330,259],{"className":10331},[285,482],[108,10333,2753],{"className":10334},[171,482],[108,10336,259],{"className":10337},[285,482],[108,10339,2753],{"className":10340},[171,482],[108,10342,259],{"className":10343},[285,482],[108,10345,2745],{"className":10346},[171,482],[108,10348,259],{"className":10349},[285,482],[108,10351,65],{"className":10352},[171,482],[108,10354,510],{"className":10355},[509],[108,10357,10359],{"className":10358},[462],[108,10360,10362],{"className":10361,"style":517},[466],[108,10363],{},[108,10365],{"className":10366},[523,977],[108,10368],{"className":10369,"style":766},[179],[108,10371,680],{"className":10372},[770],[108,10374],{"className":10375,"style":766},[179],[108,10377,10379,10382,10450,10453,10456],{"className":10378},[162],[108,10380],{"className":10381,"style":1736},[166],[108,10383,10385,10388,10447],{"className":10384},[171],[108,10386],{"className":10387},[443,977],[108,10389,10391],{"className":10390},[411],[108,10392,10394,10439],{"className":10393},[457,458],[108,10395,10397,10436],{"className":10396},[462],[108,10398,10400,10414,10422],{"className":10399,"style":1663},[466],[108,10401,10402,10405],{"style":993},[108,10403],{"className":10404,"style":997},[474],[108,10406,10408],{"className":10407},[479,480,481,482],[108,10409,10411],{"className":10410},[171,482],[108,10412,2748],{"className":10413},[171,482],[108,10415,10416,10419],{"style":1031},[108,10417],{"className":10418,"style":997},[474],[108,10420],{"className":10421,"style":1039},[1038],[108,10423,10424,10427],{"style":1042},[108,10425],{"className":10426,"style":997},[474],[108,10428,10430],{"className":10429},[479,480,481,482],[108,10431,10433],{"className":10432},[171,482],[108,10434,2748],{"className":10435},[171,482],[108,10437,510],{"className":10438},[509],[108,10440,10442],{"className":10441},[462],[108,10443,10445],{"className":10444,"style":517},[466],[108,10446],{},[108,10448],{"className":10449},[523,977],[108,10451],{"className":10452,"style":766},[179],[108,10454,680],{"className":10455},[770],[108,10457],{"className":10458,"style":766},[179],[108,10460,10462,10465],{"className":10461},[162],[108,10463],{"className":10464,"style":237},[166],[108,10466,65],{"className":10467},[171],[15,10469,10470,10471,10530,10531,10559,10560,812],{},"Each number in row 5 divided by ",[108,10472,10474,10492],{"className":10473},[111],[108,10475,10477],{"className":10476},[115],[117,10478,10479],{"xmlns":119},[121,10480,10481,10489],{},[124,10482,10483],{},[2505,10484,10485,10487],{},[127,10486,675],{},[127,10488,2745],{},[151,10490,10491],{"encoding":153},"2^5",[108,10493,10495],{"className":10494,"ariaHidden":132},[158],[108,10496,10498,10501],{"className":10497},[162],[108,10499],{"className":10500,"style":2811},[166],[108,10502,10504,10507],{"className":10503},[171],[108,10505,675],{"className":10506},[171],[108,10508,10510],{"className":10509},[2533],[108,10511,10513],{"className":10512},[457],[108,10514,10516],{"className":10515},[462],[108,10517,10519],{"className":10518,"style":2811},[466],[108,10520,10521,10524],{"style":2545},[108,10522],{"className":10523,"style":475},[474],[108,10525,10527],{"className":10526},[479,480,481,482],[108,10528,2745],{"className":10529},[171,482]," gives us a probability, and together they sum to ",[108,10532,10534,10547],{"className":10533},[111],[108,10535,10537],{"className":10536},[115],[117,10538,10539],{"xmlns":119},[121,10540,10541,10545],{},[124,10542,10543],{},[127,10544,65],{},[151,10546,65],{"encoding":153},[108,10548,10550],{"className":10549,"ariaHidden":132},[158],[108,10551,10553,10556],{"className":10552},[162],[108,10554],{"className":10555,"style":237},[166],[108,10557,65],{"className":10558},[171],". The triangle, combinatorics, and algebra are not three different stories: ",[27,10561,10562],{},"they are the same story told in three different languages",[19,10564,10566],{"id":10565},"final-reflection","Final reflection",[15,10568,10569],{},"Throughout this article we have followed a thread that connects three branches of mathematics through a single object:",[6323,10571,10572,10577,10583],{},[60,10573,10574,10576],{},[27,10575,2267],{}," tells us how many ways we can arrange a set of outcomes.",[60,10578,10579,10580,10582],{},"The ",[27,10581,6222],{}," of the binomial generates those same quantities as coefficients in a formula.",[60,10584,10585,10587],{},[27,10586,3229],{}," transforms those numbers into predictions about the real world.",[15,10589,10590,10591,10593],{},"And ",[27,10592,29],{}," is the object that unites them: a visual table where all three disciplines converge.",[15,10595,10596],{},"What started as a triangle for counting combinations — drawn in 11th-century China — ends up describing with precision what happens when you flip a coin five, ten, or a hundred times. An abstract structure created for counting objects turns out to describe random phenomena.",[15,10598,10599],{},"And there is one more detail worth mentioning. As the number of flips grows, the rows of the triangle increasingly resemble the Gaussian bell curve: that ubiquitous curve that appears in statistics, physics, and biology. The deep reason for that convergence is another story, but the fact that a triangle built from such simple sums contains the seed of one of the most important distributions in science is, at the very least, beautiful.",[15,10601,10602],{},"Mathematics has a unique quality: it reveals hidden patterns behind what appears to be pure randomness. A single coin flip is unpredictable. Ten flips already hint at an order. A hundred flips draw an almost perfect curve. And it was all written, from the very beginning, in a triangle.",[15,10604,10605,10606,10611],{},"Put theory into practice: ",[2764,10607,10610],{"href":10608,"rel":10609},"https:\u002F\u002Fflipthecoin.app\u002Fplay\u002F",[2768],"flip a coin online"," and watch combinatorics in action.",{"title":90,"searchDepth":10613,"depth":10613,"links":10614},2,[10615,10616,10620,10621,10624,10625,10628],{"id":21,"depth":10613,"text":22},{"id":51,"depth":10613,"text":52,"children":10617},[10618],{"id":382,"depth":10619,"text":383},3,{"id":2261,"depth":10613,"text":2262},{"id":2772,"depth":10613,"text":2773,"children":10622},[10623],{"id":3223,"depth":10619,"text":3224},{"id":4935,"depth":10613,"text":4936},{"id":6215,"depth":10613,"text":6216,"children":10626},[10627],{"id":8474,"depth":10619,"text":8475},{"id":10565,"depth":10613,"text":10566},"2026-05-24T10:00:00Z","Discover how Pascal's triangle connects combinatorics, algebra, and probability — explained step by step through coin flips.","md",{"excerpt":10633},{"type":8,"value":10634},[10635,10637],[11,10636,5],{"id":13},[15,10638,17],{},true,"\u002Fblog\u002Fpascals-triangle-and-probability.en",{"title":5,"description":10630},"pascals-triangle-combinatorics-and-coin-flips","blog\u002Fpascals-triangle-and-probability.en","j8G798_RqtDp5dnmnCaRWb68cHrLVS-vn2JgwDOZJGU",{"en":10642,"es":10646},"triangulo-de-tartaglia-combinatoria-y-probabilidad",{"en":10642,"es":10646},1779926069290]