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Math Help - Generating Function Application Question

  1. #1
    Junior Member
    Joined
    Sep 2009
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    26

    Generating Function Application Question

    Alan and Bob have stolen a total of $10,000 from the bank, in the form of fifty $20 bills, forty $50 bills, and seventy $100 bills. How many ways are there to split the loot evenly between them? Express your answer as a coefficient of a generating function.

    I'm having trouble solving this cause I am not sure how to fit all the conditions into a generating function.

    Any tips?

    Thanks
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  2. #2
    Senior Member
    Joined
    Dec 2010
    Posts
    470
    The coefficient of x^{5000}y^{5000} in the expression:


    <br />
(\sum_{i=0}^{50}(x^{20})^i(y^{20})^{50-i})<br />
(\sum_{i=0}^{40}(x^{50})^i(y^{50})^{40-i})<br />
(\sum_{i=0}^{70}(x^{100})^i(y^{100})^{70-i})<br />

    Power of x represents money given to Alan, and power of y represents money given to Bob.
    I think you can see how all the numbers fit together.

    Definitely wouldn't want to calculate this by hand
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