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.