you volunteer to help drive children at a charity event to the zoo, but you can fit only 7 of the 15…

you volunteer to help drive children at a charity event to the zoo, but you can fit only 7 of the 15 children present in your van. how many different groups of 7 children can you drive? how many different groups of 7 children can you drive?

you volunteer to help drive children at a charity event to the zoo, but you can fit only 7 of the 15 children present in your van. how many different groups of 7 children can you drive? how many different groups of 7 children can you drive?

Answer

Explanation:

Step1: Identificar el problema de combinatoria

Este es un problema de combinaciones donde se deben calcular el número de grupos de 7 elementos que se pueden formar a partir de 15 elementos. La fórmula de combinaciones es $C(n,r)=\frac{n!}{r!(n - r)!}$, donde $n$ es el número total de elementos y $r$ es el número de elementos en cada grupo. Aquí, $n = 15$ y $r=7$.

Step2: Calcular los factoriales

$n!=15! = 15\times14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, $r!=7!=7\times6\times5\times4\times3\times2\times1$, y $(n - r)!=(15 - 7)!=8!=8\times7\times6\times5\times4\times3\times2\times1$. $C(15,7)=\frac{15!}{7!(15 - 7)!}=\frac{15!}{7!8!}=\frac{15\times14\times13\times12\times11\times10\times9\times8!}{7\times6\times5\times4\times3\times2\times1\times8!}$

Step3: Simplificar la expresión

$\frac{15\times14\times13\times12\times11\times10\times9}{7\times6\times5\times4\times3\times2\times1}=\frac{15\times14\times13\times12\times11\times10\times9}{5040}$ $15\times14\times13\times12\times11\times10\times9=32432400$ $\frac{32432400}{5040}=6435$

Answer:

6435