Please answer the following question completely and fully: I have posted the que
ID: 3405672 • Letter: P
Question
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?
Explanation / Answer
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