A club has 24 members: 3 freshmen, 6 sophomore, 10 juniors, 5 seniors.
a) How many ways are there to choose a committee of four people with one person designated as the committee chair?
b) How many ways are there to choose a committee of five people with two juniors who serve as the chair and vice chair and three at-large members who are neither juniors nor seniors?
Any help would be greatly appreciated.
For a) I did C(24, 4) * C(4,1) but not sure if that is correct or not. I assumed out of 24 you choose 4 but then 1 of those 4 is the chair.
For b) I did C(10,2) * C(9,3) but I'm pretty sure that is wrong. I got the 10,2 because you pick 2 juniors out of the 10, and the 9,3 because you pick three out of the freshmen + sophomore
Best Answer
(a) There are 24 ways to pick a chair. After choosing a chair, there are 23 choices left to choose the other three. Therefore, there are $$\binom{24}{1}\times \binom{23}{3} = \binom{24}{4}\times\binom{4}{1}= 42504$$ ways to choose that committee.
(b) Since chair is different from a vice chair, you should consider them separately. The order matters, meaning, choose a junior for the chair, another junior for the vice chair. If you just choose two juniors, that doesn't tell you which chair that person is because there are two ways to do that. After, choose three non-juniors/non-senoirs. Since their titles don't matter, the order doesn't matter. You only need 3. Therefore, there are $$2\binom{10}{2}\binom{9}{3} = \binom{10}{1}\binom{9}{1}\binom{9}{3} = 7560$$ ways to choose this committee.