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.

Printable View

- Feb 7th 2010, 12:19 PMCoraGBgenerating functionIf 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.

- Feb 7th 2010, 12:29 PMMoo
Hello,

Find the expansion of the generating function, and the answer you're looking for will be the coefficient in front of $\displaystyle x^m$ - Feb 7th 2010, 12:52 PMCoraGB
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?

- Feb 8th 2010, 07:33 AMawkward
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 $\displaystyle f(x) = 3$.

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

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

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