A box of candy hearts contains 52 hearts of which 19 are whi

A box of candy hearts contains 52 hearts, of which 19 are white, 10 are tan, 7 are pink, 3 are purple, 5 are yellow, 2 are orange, and 6 are green. If you select nine pieces of candy randomly from the box, without replacement, give the probability that
(a) Three of the hearts are white.
(b) Three are white, two are tan, one is pink, one is yellow, and two are green.

Solution

a) Assuming that we want EXACTLY three white hearts (and wishing whoever assigned you this problem would state it clearly!) there are
52C9 ways to select 9 hearts,
19C3 ways to select 3 white hearts, and
33C6 ways to select 6 non-white hearts.

Remembering that nCr = n! / [(n-4)! r!], the probability is
19C3 * 33C6 / 52C9
and I\'m going to leave the arithmetic to you.

b) Expanding on the same idea, you can see that the answer to this will be
19C3 * 10C2 * 7C1 * 5C1 * 6C2 / 52C9
and again, I\'ll leave the arithmetic to you.