Question

you volunteer to help drive children at a charity event to the zoo, but you can fit only 6 of the 17 children present in your van. how many different groups of 6 children can you drive? how many different groups of 6 children can you drive?

Explanation:

Step1: Identify combination formula

The problem is about combinations. The formula for combinations is $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n$ is the total number of items, and $r$ is the number of items to be chosen. Here, $n = 17$ and $r=6$.

Step2: Calculate factorial values

$n!=17! = 17\times16\times15\times14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, $r!=6!=6\times5\times4\times3\times2\times1$, and $(n - r)!=(17 - 6)!=11!=11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$. Then $C(17,6)=\frac{17!}{6!(17 - 6)!}=\frac{17!}{6!×11!}=\frac{17\times16\times15\times14\times13\times12\times11!}{6\times5\times4\times3\times2\times1\times11!}$.

Step3: Simplify the expression

Cancel out the $11!$ terms. Then we have $\frac{17\times16\times15\times14\times13\times12}{6\times5\times4\times3\times2\times1}$. $17\times16\times15\times14\times13\times12=8910720$ and $6\times5\times4\times3\times2\times1 = 720$. $\frac{8910720}{720}=12376$.

Answer:

$12376$