# How many 5-card hands from a standard 52-card deck contain exactly 1 king and exactly 1 heart?

Is my solution correct?

There are two cases:

Case 1: (the chosen heart is not a king) So there are $$3 \choose 1$$ ways to choose the king, and $$12 \choose 1$$ ways to choose the heart, and then $$52-12-3 \choose 3$$ ways to choose rest of the three cards.

Case 2: (the chosen heart is a king) So there are $$1 \choose 1$$ way to choose the king (heart) card, and then since rest of the 4 cards cannot be a king nor heart, there are $$52-1-12-3 \choose 4$$ ways to choose the rest.

Then adding these two cases together we get the result.

In the first case, once you select a king and a heart that is not the king of hearts, there are $$16$$ cards from which you cannot select the other three cards: the four kings and the hearts, as $$13 + 4 - 1 = 16$$. Therefore, you should have $$\binom{3}{1}\binom{12}{1}\binom{52 - 13 - 4 + 1}{3}$$