# Thread: generating function

1. ## generating function

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.

2. Hello,

Find the expansion of the generating function, and the answer you're looking for will be the coefficient in front of $x^m$

3. I have seen an example with two decks where the binomial theorem was used twice, is it possible to use it three times, if so how?

4. Originally Posted by CoraGB
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.

Hi CoraGB,

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 $f(x) = 3$.

(2) Let $a_m$ be the number of ways to choose a hand of m cards from a deck of size n, and let $g(x) = \sum a_m x^m$. Then $g(x) = (1+x)^n$.

Finally, the OPSGF of the number of ways to choose the deck and choose a hand is just

$f(x) \cdot g(x) = 3 (1+x)^n$.