Question

How many different committees can be formed from 12 teachers and 35 students if the committee consists of 4 teachers and 3 students?

Answer 1

Solution:

Given that a committee of 4 teachers and 3 students is to be formed from 12 teachers and 35 students, we have

$12C4\text{ }\times35C3$

Recall that:

$nCr=\frac{n!}{(n-r)!\times r!}$

This gives:

$\begin{gathered} \frac{12!}{(12-4)!\times4!}\times\frac{35!}{(35-3)!\times3!} \\ =\frac{12!}{8!\times4!}\times\frac{35!}{32!\times3!} \\ =\frac{12\times11\times10\times9\times\cancel8!}{\cancel8!\times4\times3\times2\times1}\times\frac{35\times34\times33\times\cancel32!}{\cancel32!\times3\times2\times1} \\ =3239775 \end{gathered}$

Hence, the number of committees that can be formed is

$\begin{equation*} 3239775 \end{equation*}$