# Choosing n balls from 2 types

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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## 1 Answer

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