[reuben_tate]

Although this problem is already solved, here is an alternative solution which might be helpful for solving future similar problems.

So the probability of getting X hearts in a hand of 5 cards is equal to:

The number of possible hands with X hearts
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The number of possible hands overall

The number of possible hands is simple, you have 52 cards and you have to pick 5, so there are C(52,5) such possible hands.

Now consider the number of possible hands with X hearts. There are 13 hearts in the deck and 52-13 non-hearts in the deck. Thus, you must pick X hearts from 13 possible hearts and 5-X hearts from the 52-13 non-hearts, i.e., there are C(13,X) ways to pick out the hearts for the hand and C(52-13, 5-X) ways of picking the non-hearts for the hand. So together, there are C(13,X) * C(52-13, 5-x) ways of forming a 5 card hand with X hearts.

So, using the formula above, the probability of getting a hand with X hearts is:

C(13,X) * C(52-13, 5-X)
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C(52,5)