82:@0.067867:0.963104:0.095634:0.963104:0.095634:0.941283:0.067867:0.941283:0.000000:0.000000 Matemática:@0.120102:0.956142:0.204868:0.956142:0.204868:0.943479:0.120102:0.943479:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 9.º EGB:@0.120102:0.967458:0.170641:0.967458:0.170641:0.954794:0.120102:0.954794:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 DUA:@0.070720:0.123711:0.114363:0.123711:0.114363:0.103460:0.070720:0.103460:0.000000:0.000000:0.000000 . Representación:@0.114363:0.123082:0.246653:0.123082:0.246653:0.106182:0.114363:0.106182:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Triángulo de Pascal. :@0.286308:0.063945:0.641405:0.063945:0.641405:0.027071:0.286308:0.027071:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Teorema del binomio:@0.286308:0.094120:0.658050:0.094120:0.658050:0.057246:0.286308:0.057246:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.094760:0.108604:0.183816:0.108604:0.183816:0.011516:0.094760:0.011516:0.000000:0.000000 El teorema del binomio se refiere a la suma o diferencia de dos cantidades elevadas :@0.287495:0.153866:0.907051:0.153866:0.907051:0.137477:0.287495:0.137477:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a una potencia entera. Es de mucha utilidad para calcular probabilidades, resolver :@0.287495:0.170462:0.907125:0.170462:0.907125:0.154073:0.287495:0.154073:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 problemas de combinatoria, desarrollar potencias de binomios y calcular coeficientes :@0.287495:0.187059:0.907125:0.187059:0.907125:0.170670:0.287495:0.170670:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 binomiales que se verán en temas posteriores.:@0.287495:0.203655:0.626079:0.203655:0.626079:0.187266:0.287495:0.187266:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Triángulo de Pascal:@0.287495:0.236413:0.482578:0.236413:0.482578:0.214903:0.287495:0.214903:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ¿Qué estructura tiene el triángulo denominado triángulo de Pascal que se observa :@0.287495:0.259194:0.907160:0.259194:0.907160:0.242805:0.287495:0.242805:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 en la imagen?:@0.287495:0.275790:0.389824:0.275790:0.389824:0.259401:0.287495:0.259401:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Se trata de un triángulo simétrico de números enteros. Está conformado por filas :@0.287495:0.299509:0.907029:0.299509:0.907029:0.283120:0.287495:0.283120:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 que tienen el número 1 al inicio y al final de cada fila. Empieza con un 1 en la primera :@0.287495:0.316106:0.907244:0.316106:0.907244:0.299717:0.287495:0.299717:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 fila, y en las filas siguientes muestra números de forma que cada uno de ellos son la :@0.287495:0.332702:0.907141:0.332702:0.907141:0.316313:0.287495:0.316313:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 suma de los dos números que tiene encima. :@0.287495:0.349298:0.615301:0.349298:0.615301:0.332909:0.287495:0.332909:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejemplo::@0.287495:0.617399:0.350979:0.617399:0.350979:0.600761:0.287495:0.600761:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Obtén las filas 9 y 10 del triángulo de Pascal.:@0.287495:0.641118:0.608940:0.641118:0.608940:0.624729:0.287495:0.624729:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Solución:@0.287495:0.668405:0.351273:0.668405:0.351273:0.651767:0.287495:0.651767:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Para obtener la fila 9, sumamos los números de la fila 8.:@0.287495:0.692124:0.689378:0.692124:0.689378:0.675735:0.287495:0.675735:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Para formar la fila 10, sumamos los obtenidos de la fila 9.:@0.287495:0.817786:0.697763:0.817786:0.697763:0.801397:0.287495:0.801397:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 8 28 56 70 56 28 8 1:@0.370123:0.728733:0.844170:0.728733:0.844170:0.710312:0.370123:0.710312:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 9 36 84 126 126 84 36 9 1:@0.344628:0.778421:0.866654:0.778421:0.866654:0.760000:0.344628:0.760000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 9 36 84 126 126 84 36 9 1:@0.325178:0.847058:0.883218:0.847058:0.883218:0.828637:0.325178:0.828637:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 10 45 120 210 252 210 120 45 10 1:@0.306716:0.896745:0.901700:0.896745:0.901700:0.878324:0.306716:0.878324:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 :@0.337459:0.567915:0.350141:0.567915:0.350141:0.552337:0.337459:0.552337:0.000000:0.000000 :@0.350141:0.567915:0.444038:0.567915:0.444038:0.552337:0.350141:0.552337:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 8:@0.394133:0.567915:0.403430:0.567915:0.403430:0.552337:0.394133:0.552337:0.000000 28 :@0.444038:0.567915:0.472785:0.567915:0.472785:0.552337:0.444038:0.552337:0.000000:0.000000:0.000000:0.000000:0.000000 56 70 56 :@0.472785:0.567915:0.714690:0.567915:0.714690:0.552337:0.472785:0.552337:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 28:@0.714690:0.567915:0.733285:0.567915:0.733285:0.552337:0.714690:0.552337:0.000000:0.000000 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6:@0.461745:0.517309:0.471043:0.517309:0.471043:0.501731:0.461745:0.501731:0.000000 15:@0.518419:0.517309:0.537014:0.517309:0.537014:0.501731:0.518419:0.501731:0.000000:0.000000 20 :@0.537014:0.517309:0.650361:0.517309:0.650361:0.501731:0.537014:0.501731:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 15:@0.650361:0.517309:0.668956:0.517309:0.668956:0.501731:0.650361:0.501731:0.000000:0.000000 :@0.668956:0.517309:0.776389:0.517309:0.776389:0.501731:0.668956:0.501731:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 6:@0.716332:0.517309:0.725629:0.517309:0.725629:0.501731:0.716332:0.501731:0.000000 1:@0.776389:0.517309:0.785687:0.517309:0.785687:0.501731:0.776389:0.501731:0.000000 1:@0.468916:0.467960:0.478213:0.467960:0.478213:0.452382:0.468916:0.452382:0.000000 :@0.478213:0.467960:0.709145:0.467960:0.709145:0.452382:0.478213:0.452382:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4:@0.532357:0.467960:0.541654:0.467960:0.541654:0.452382:0.532357:0.452382:0.000000 6:@0.589030:0.467960:0.598328:0.467960:0.598328:0.452382:0.589030:0.452382:0.000000 4:@0.645704:0.467960:0.655001:0.467960:0.655001:0.452382:0.645704:0.452382:0.000000 1:@0.709145:0.467960:0.718442:0.467960:0.718442:0.452382:0.709145:0.452382:0.000000 1:@0.528973:0.422471:0.538270:0.422471:0.538270:0.406893:0.528973:0.406893:0.000000 :@0.538270:0.422471:0.649088:0.422471:0.649088:0.406893:0.538270:0.406893:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.592414:0.422471:0.601712:0.422471:0.601712:0.406893:0.592414:0.406893:0.000000 1:@0.649088:0.422471:0.658385:0.422471:0.658385:0.406893:0.649088:0.406893:0.000000 1:@0.557301:0.399726:0.566599:0.399726:0.566599:0.384148:0.557301:0.384148:0.000000 :@0.566599:0.399726:0.617358:0.399726:0.617358:0.384148:0.566599:0.384148:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.617358:0.399726:0.630040:0.399726:0.630040:0.384148:0.617358:0.384148:0.000000:0.000000 1:@0.493860:0.445869:0.503157:0.445869:0.503157:0.430291:0.493860:0.430291:0.000000 :@0.503157:0.445869:0.684184:0.445869:0.684184:0.430291:0.503157:0.430291:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3:@0.557301:0.445869:0.566599:0.445869:0.566599:0.430291:0.557301:0.430291:0.000000 3:@0.617358:0.445869:0.630040:0.445869:0.630040:0.430291:0.617358:0.430291:0.000000:0.000000 1:@0.684184:0.445869:0.693481:0.445869:0.693481:0.430291:0.684184:0.430291:0.000000 1:@0.429832:0.493068:0.439130:0.493068:0.439130:0.477490:0.429832:0.477490:0.000000 :@0.439130:0.493068:0.549176:0.493068:0.549176:0.477490:0.439130:0.477490:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.486120:0.493068:0.498802:0.493068:0.498802:0.477490:0.486120:0.477490:0.000000:0.000000 10 :@0.548925:0.493068:0.570904:0.493068:0.570904:0.477490:0.548925:0.477490:0.000000:0.000000:0.000000 :@0.570904:0.493068:0.621279:0.493068:0.621279:0.477490:0.570904:0.477490:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 10:@0.621027:0.493068:0.639622:0.493068:0.639622:0.477490:0.621027:0.477490:0.000000:0.000000 :@0.639622:0.493068:0.746285:0.493068:0.746285:0.477490:0.639622:0.477490:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.686613:0.493068:0.695910:0.493068:0.695910:0.477490:0.686613:0.477490:0.000000 1:@0.746034:0.493068:0.755331:0.493068:0.755331:0.477490:0.746034:0.477490:0.000000 1:@0.370629:0.543348:0.379927:0.543348:0.379927:0.527770:0.370629:0.527770:0.000000 :@0.379927:0.543348:0.487794:0.543348:0.487794:0.527770:0.379927:0.527770:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 7:@0.426029:0.543348:0.435327:0.543348:0.435327:0.527770:0.426029:0.527770:0.000000 21:@0.487291:0.543348:0.505886:0.543348:0.505886:0.527770:0.487291:0.527770:0.000000:0.000000 35 35 :@0.505886:0.543348:0.685640:0.543348:0.685640:0.527770:0.505886:0.527770:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 21:@0.685137:0.543348:0.703732:0.543348:0.703732:0.527770:0.685137:0.527770:0.000000:0.000000 :@0.703732:0.543348:0.805738:0.543348:0.805738:0.527770:0.703732:0.527770:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 7:@0.749835:0.543348:0.759132:0.543348:0.759132:0.527770:0.749835:0.527770:0.000000 1:@0.805235:0.543348:0.814533:0.543348:0.814533:0.527770:0.805235:0.527770:0.000000 1:@0.589013:0.378239:0.598311:0.378239:0.598311:0.362661:0.589013:0.362661:0.000000 . Representación:@0.114363:0.123082:0.246653:0.123082:0.246653:0.106182:0.114363:0.106182:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Matemática e historia:@0.088760:0.423400:0.244757:0.423400:0.244757:0.408036:0.088760:0.408036:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Blaise Pascal, con su :@0.112632:0.442046:0.248426:0.442046:0.248426:0.427147:0.112632:0.427147:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 análisis del triángulo que :@0.078826:0.457133:0.248426:0.457133:0.248426:0.442234:0.078826:0.442234:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 lleva su apellido, :@0.135583:0.472221:0.248426:0.472221:0.248426:0.457322:0.135583:0.457322:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 contribuyó a la :@0.145249:0.487308:0.248427:0.487308:0.248427:0.472409:0.145249:0.472409:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 conformación de teorías :@0.081992:0.502396:0.248426:0.502396:0.248426:0.487497:0.081992:0.487497:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemáticas, como :@0.113487:0.517484:0.248426:0.517484:0.248426:0.502585:0.113487:0.502585:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 también lo hicieron los :@0.091993:0.532571:0.248426:0.532571:0.248426:0.517672:0.091993:0.517672:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 trabajos del ecuatoriano :@0.083165:0.547659:0.248424:0.547659:0.248424:0.532760:0.083165:0.532760:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de origen alemán, :@0.123991:0.562747:0.248427:0.562747:0.248427:0.547848:0.123991:0.547848:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Peter Thullen.:@0.154832:0.577834:0.244759:0.577834:0.244759:0.562935:0.154832:0.562935:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Interdisciplinariedad:@0.090129:0.394241:0.247335:0.394241:0.247335:0.377340:0.090129:0.377340:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ©:@0.046636:0.284435:0.046636:0.279328:0.034554:0.279328:0.034554:0.284435:0.000000 Shutterstock:@0.046636:0.279328:0.046636:0.241610:0.034725:0.241610:0.034725:0.279328:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 En el triángulo de Pascal :@0.083199:0.695149:0.248426:0.695149:0.248426:0.680250:0.083199:0.680250:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 se observan algunas :@0.108914:0.710237:0.248426:0.710237:0.248426:0.695338:0.108914:0.695338:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 particularidades. Por :@0.109333:0.725324:0.248426:0.725324:0.248426:0.710425:0.109333:0.710425:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ejemplo, al sumar los :@0.104592:0.740412:0.248426:0.740412:0.248426:0.725513:0.104592:0.725513:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 números de la fila, se :@0.105714:0.755500:0.248428:0.755500:0.248428:0.740601:0.105714:0.740601:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 obtienen las potencias :@0.093804:0.770587:0.248426:0.770587:0.248426:0.755688:0.093804:0.755688:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de 2.:@0.212762:0.785675:0.244759:0.785675:0.244759:0.770776:0.212762:0.770776:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.141344:0.805775:0.146332:0.805775:0.146332:0.796958:0.141344:0.796958:0.000000 1:@0.224247:0.805775:0.229235:0.805775:0.229235:0.796958:0.224247:0.796958:0.000000 2:@0.223431:0.816858:0.228418:0.816858:0.228418:0.808041:0.223431:0.808041:0.000000 4:@0.223431:0.827970:0.228418:0.827970:0.228418:0.819153:0.223431:0.819153:0.000000 8:@0.223431:0.838957:0.228418:0.838957:0.228418:0.830141:0.223431:0.830141:0.000000 16:@0.220932:0.849901:0.230907:0.849901:0.230907:0.841085:0.220932:0.841085:0.000000:0.000000 32:@0.220932:0.860779:0.230907:0.860779:0.230907:0.851962:0.220932:0.851962:0.000000:0.000000 64:@0.220932:0.871686:0.230907:0.871686:0.230907:0.862869:0.220932:0.862869:0.000000:0.000000 128:@0.221963:0.882490:0.236925:0.882490:0.236925:0.873673:0.221963:0.873673:0.000000:0.000000:0.000000 1:@0.148519:0.816858:0.153507:0.816858:0.153507:0.808041:0.148519:0.808041:0.000000 1:@0.155704:0.827970:0.160692:0.827970:0.160692:0.819153:0.155704:0.819153:0.000000 1:@0.162889:0.838957:0.167877:0.838957:0.167877:0.830141:0.162889:0.830141:0.000000 1:@0.170074:0.849901:0.175062:0.849901:0.175062:0.841085:0.170074:0.841085:0.000000 1:@0.177259:0.860779:0.182246:0.860779:0.182246:0.851962:0.177259:0.851962:0.000000 1:@0.184444:0.871686:0.189431:0.871686:0.189431:0.862869:0.184444:0.862869:0.000000 1:@0.191628:0.882490:0.196616:0.882490:0.196616:0.873673:0.191628:0.873673:0.000000 1 7 21 35 35 21 7:@0.091021:0.882490:0.182237:0.882490:0.182237:0.873673:0.091021:0.873673:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 6 15:@0.098206:0.871686:0.134422:0.871686:0.134422:0.862869:0.098206:0.862869:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 5 10:@0.105614:0.860779:0.141616:0.860779:0.141616:0.851962:0.105614:0.851962:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 4:@0.112575:0.849901:0.131933:0.849901:0.131933:0.841085:0.112575:0.841085:0.000000:0.000000:0.000000 1 3 3:@0.119760:0.838957:0.153487:0.838957:0.153487:0.830141:0.119760:0.830141:0.000000:0.000000:0.000000:0.000000:0.000000 1 2:@0.126945:0.827970:0.146303:0.827970:0.146303:0.819153:0.126945:0.819153:0.000000:0.000000:0.000000 1:@0.134130:0.816858:0.139118:0.816858:0.139118:0.808041:0.134130:0.808041:0.000000 4:@0.155685:0.849901:0.160672:0.849901:0.160672:0.841085:0.155685:0.841085:0.000000 6:@0.141315:0.849901:0.146303:0.849901:0.146303:0.841085:0.141315:0.841085:0.000000 6:@0.170064:0.871686:0.175052:0.871686:0.175052:0.862869:0.170064:0.862869:0.000000 5:@0.162879:0.860779:0.167867:0.860779:0.167867:0.851962:0.162879:0.851962:0.000000 10:@0.146011:0.860779:0.155986:0.860779:0.155986:0.851962:0.146011:0.851962:0.000000:0.000000 20 15:@0.138826:0.871686:0.163171:0.871686:0.163171:0.862869:0.138826:0.862869:0.000000:0.000000:0.000000:0.000000:0.000000 Matemática:@0.115799:0.662707:0.223340:0.662707:0.223340:0.645806:0.115799:0.645806:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 M.4.1.33. :@0.290067:0.965521:0.338404:0.965521:0.338404:0.954766:0.290067:0.954766:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Reconocer y calcular productos notables e identificar factores de expresiones algebraicas.:@0.338404:0.965521:0.755012:0.965521:0.755012:0.955091:0.338404:0.955091:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.568842:0.965521:0.571410:0.965521:0.571410:0.955091:0.568842:0.955091:0.000000