Question

you volunteer to help drive children at a charity event to the zoo, but you can fit only 6 of the 15 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 the problem type

This is a combination problem since the order of the children in the group does not matter. The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 15 \) (total number of children) and \( k = 6 \) (number of children in each group).

Step2: Calculate factorials

First, calculate \( n!=15! = 15\times14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1 \), \( k!=6! = 6\times5\times4\times3\times2\times1 \), and \( (n - k)!=(15 - 6)!=9! = 9\times8\times7\times6\times5\times4\times3\times2\times1 \).

Step3: Substitute into the combination formula

\( C(15, 6)=\frac{15!}{6!(15 - 6)!}=\frac{15\times14\times13\times12\times11\times10\times9!}{6!\times9!} \). The \( 9! \) terms cancel out. Then, \( 6! = 720 \), and \( 15\times14\times13\times12\times11\times10=\ 15\times14 = 210\), \( 210\times13 = 2730\), \( 2730\times12 = 32760\), \( 32760\times11 = 360360\), \( 360360\times10 = 3603600 \). Now, divide by \( 720 \): \( \frac{3603600}{720}=5005 \).

Answer:

\( 5005 \)