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How many ways are there to split $6$ copies of one book, $7$ copies of a second book, and $11$ copies of a third book between two teachers if each teacher gets $12$ books and each teacher gets at least $2$ copies of each book?

I have the following generating function $$ (x^2 + x^3 + \ldots + x^6)\cdot(x^2 + x^3 + \ldots + x^7)\cdot(x^2 + x^3 + \ldots + x^{11}) $$

I also used identities to express that in a more concise way, but didn't get me anywhere.

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Once you decide what the 1st teacher gets, you're done.

The 1st teacher gets at least 2 of 1st book, but also at most 4 (since 2nd teacher must get at least 2).

So the generating function you need is $$(x^2+x^3+x^4)(x^2+\cdots+x^5)(x^2+\cdots+x^9)$$ and you want the coefficient of $x^{12}$ in this.

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