In how many ways a commitee of three people

Question

A committee of three people is to be selected from four women and five men. The rules state that there must be at least one man and one woman on the committee. In how many ways can the committee be chosen? Subsequently one of the men and one of the women marry each other. The rules also state that a married couple may not both serve on the committee. In how many ways can the committee be chosen now?

Answer 1

Some hints.

For the first part, split it into committees with two women, and committees with two men. These groups are distinct, and easier to count.

Once you have this, to solve the second part, remove the ones that have the married couple.

Answer 2

For part A we have by inclusion exclusion principle the answer as $$\binom {9}{3}- \binom {4}{3} - \binom {5}{3}$$

For part second subtract the cases in which both the partners of the couple are in the committee. Hence the number of ways to select three people for the committee satisfying given condition can be given by inclusion exclusion principle as $$\binom {9}{3}- \binom {4}{3} - \binom {5}{3}- \binom {7}{1}$$ Note: For the second part I have considered the question to be asking for the ways to form a committee satisfying both of the given conditions.