$$\binom{3}{2}\binom{7}{3}\binom{22}{0}+\binom{1}{1}\binom{3}{1}\binom{7}{2}\binom{21}{1}=3\times35\times1+1\times3\times21\times21=1428$$
Do you see why?
The first term deals with the case that $2$ kings are selected from the $3$ non-heart kings, $3$ hearts from the $7$ non-king hearts and $0$ from the rest.
The second term: $1$ heart-king from $1$ heart-king, $1$ king of $3$ nonheart-kings, $2$ hearts from $7$ nonking-hearts and $1$ from the rest.