Question

A committee of 5 people is to be chosen from a group of 12 people, 6 women and 6 men. In how any ways can the committee be chosen as to include exactly 3 men?

Answer 1

In 300 ways the committee can be chosen as to include exactly 3 men

Step-by-step explanation:

The  general formula is $C_{n,k}=\frac{n!}{(k!)(n-k)!}$

If we choose exactly 3 men, the remain 2 people of the committee must be women. So the group of 5 people will be having exactly 3 men and 2 women

The possible ways of selecting exactly 3 men is

$C_{6,3} \cdot C_{6,2} = \frac{6!}{3!(6-3)!} \cdot \frac{6!}{2!(6-2)!}$

$C_{6,3} \cdot C_{6,2} = \frac{6!}{3!(3)!} \cdot \frac{6!}{2!(4)!}$

$C_{6,3} \cdot C_{6,2} = \frac{720}{(6)(6)} \cdot \frac{720}{2(24)}$

$C_{6,3} \cdot C_{6,2} = \frac{720}{(36)} \cdot \frac{720}{48}$

$C_{6,3} \cdot C_{6,2} = 20 \cdot 15$ =  300 ways