If you have a three decks of cards, how many ways are there to pick a hand of m cards from n distinct cards in a single deck. Use a generating function.
I have Gn(x) = (1+x+x^2+x^3)^n but don't know where to go from here.
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