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I want to choose n balls from 2 types using generating functions.

Normally I would think to write $$f(x)=(1+x+...+x^n)^2 = \left ( \frac{1-x^{n+1}}{1-x} \right )^2$$ and then look for the coefficient of $x^n$, but I'm thinking that since any coefficient after $x^n$ won't contribute anything I should be able to use the simpler expression $$(1+x+...)^2 = \left ( \frac{1}{1-x} \right )^2$$ Is this correct? Is it something I would need to prove or is the simple explanation above sufficient?

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It is correct that the simpler expression gives the same answer as the original. As to whether you need to prove it, that would depend on your audience. If you are a 1st-year undergraduate writing a homework assignment, the marker might want to be convinced that you know what you're doing. If you are writing a paper for Inventiones Mathematicae, you can safely assume the reader will fill in the dots.

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This is part of a larger 1st year undergrad assignment. Do you think stating that since the coefficients greater than $x^n$ don't contribute they can be ignored and the simpler expression can thus be used would be a sufficient explanation? –  Robert S. Barnes Jan 20 '12 at 11:38
    
I think there is only one person who can answer that question, and she's the one who gave you the assignment. Better ask her. Alternatively, err on the safe side; you'll never get into trouble for giving too much justification. –  Gerry Myerson Jan 20 '12 at 11:59

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