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Math Help - generating function

  1. #1
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    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.

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  2. #2
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    Hello,

    Find the expansion of the generating function, and the answer you're looking for will be the coefficient in front of x^m
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    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?
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    Quote Originally Posted by CoraGB View Post
    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.
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