(Clarification help) There are 18 baseball player and 16 basketball players who are running a sports camp.

Question

There are 18 baseball player and 16 basketball players who are running a sports camp. In how many ways could they form a 4-member fund raising committee if the committee:

b) have exactly one baseball player? <- For this, I know that basketball players, you would do 16! / 13!3! = 560 (3 because we are finding of the 1 exact baseball player that we're trying to find) for basketball players because we don't care about the order so we would use the nCr = n! / (n - r)r! formula, whereas for baseball players, you do 18! / 17! = 18, we use the C(n, r) formula because we do care about the order. Multiplying 560 and 18, giving 10, 080 is the correct answer.

c) have exactly two basketball players? As I was doing the exact same for exactly two basketball players it was incorrect for some reason, I did 16! / 14! = 240, because we care about the order, for baseball players I then did 18! / 16!2! = 153, multiplied them, 240 x 153 = 36,720, but the correct answer was supposed to be 18,360. Please explain the reason for this!

d) can be all basketball or all baseball players?

Answer 1

(b) C(18,1) x C (16,3) = 18 x 560 = 10 080

(c) C (16,2) x C (18,2) = 120 x 153 = 18 360. [ the order doesnt matter for the basketball players, so 16!/14! 2! = 120]

(d) All basketball : C (16.4) x C (18,0) = 1820 X 1 = 1820

All baseball C (18,4) x C (16,0) = 3060 x 1 = 3060

So 1820 + 3060 = 4880