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Exact solution

Posted By: Jason Lee
Date: Wednesday, 25 August 2010, at 1:03 a.m.

In Response To: Exact solution (Chuck Bower)

Chuck (and other interested parties) --

f(x) is called a generating function. It is not (usually) treated as a function in the normal sense, in that you typically will not plug in values for x. Rather, it is the coefficients which encapsulate information, often counting something.

In this case, the coefficients on the monomials that make up f(x) count how many ways there are of rolling a certain number of pips. So, in using the example I set before, one of the terms in f(x) was 5x^8. That tells you there are 5 ways to roll 8 pips (26, 62, 35, 53, 22). We are defining f(x) to encapsulate this information.

Then, when we compute (f(x))^9, you get some gigantic polynomial (which I posted before). Here, the coefficient on x^n counts the number of ways of rolling a pair of differently colored dice nine times and getting a total of n pips (I won't prove this). If we both roll a set of {blue, red} dice nine times, we'll say that our sequences are the same iff we get the same rolls each time, where colors can distinguish different rolls.

One application of using generating functions with regard to dice are to find so-called Sicherman dice.

JLee

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