Question

 Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with six members if it must have the same number of men and women?

Answer 1

If it must be the same number of men and women, it should be 3 men and 3 women in the department.

So we can choose 3 men out of 10 in $C^3_{10}=\frac{10!}{3!7!}=120$ ways.

We can choose 3 women out of 15 women in $C^3_{15}=\frac{15!}{12!3!}=455$ ways.

So we can form the department in 120x455=54600 ways.