# How many 5-card hands are there with 3 hearts and a three-of-a-kind?

How many possible 5-card hands from a deck of 52 cards are there that consists of 3 hearts and a three-of-a-kind?

I did:

C(13,3) = number of ways to choose three hearts

C(3,1) = number of ways to choose which heart gets to be part of the three-of-a-kind

C(3,2) = number of ways to choose 2 cards from the remaining suits to complete the three-of-a-kind.

So, in total, we have: C(13,3) * C(3,1) * C(3,2) possible hands.

As a check, here is another approach: Select one of the thirteen ranks. Choose a heart and two of the three remaining suits of that rank to form a three-of-a-kind that includes a heart. Choose two of the remaining twelve hearts. This yields $$\binom{13}{1}\binom{3}{2}\binom{12}{2} = \binom{13}{3}\binom{3}{1}\binom{3}{2} = 2574$$