Find the expansion of the generating function, and the answer you're looking for will be the coefficient in front of
Break the problem into two parts: (1) Find the OPSGF (ordinary power series generating function) of the number of ways to choose the deck; (2) find the OPSGF of the number of hands in a deck.
(1) Is easy: it's just .
(2) Let be the number of ways to choose a hand of m cards from a deck of size n, and let . Then .
Finally, the OPSGF of the number of ways to choose the deck and choose a hand is just