Please answer the following question completely and fully I

Please answer the following question completely and fully: I have posted the question as both text and an image, please only answer the question once.

Seven women and nine men are on the faculty in the mathematics department at a school.

a) How many ways are there to select a committee of five members of the department if at least one woman must be on the committee?

b) How many ways are there to select a committee of five members of the department if at least one woman and at least one man must be on the committee?

Solution

a) There are 16 faculty members in the department, therefore C(16, 5) different combinations of 5 people are possible. However, C(9, 5) of them consist entirely of men. Therefore there are C(16, 5) C(9, 5) = 16! 5!11! 9! 5!4! ways to select a committee with at least one woman.

ncr(16,5) - ncr(9,5)= 4242

b) we also have to subtract C(7, 5) (the number of combinations that consist entirely of women). So there are C(16, 5) C(9, 5) C(7, 5) = 16! 5!11! 9! 5!4! 7! 5!2! ways to select a committee with at least one woman and at least one man.

ncr(16,5) - ncr(9,5)-ncr(7,5)= 4221