<s>[INST] Find coefficient of x in a generating function The problem is as follows:
$\text{Determine the coef. of } x^{10} \text{ in } (x^3 + x^5 + x^6)(x^4 + x^5 + x^7)(1+x^5+x^{10}+x^{15}+...)$

I factored out some $x$'s, to get $x^3(1+x^2+x^3)x^4(1+x+x^3)(1+x^5+x^{10}+x^{15}+...)$and then combined the factored terms to get  $x^7(1+x^2+x^4)(1+x+x^3)(1+x^5+x^{10}+x^{15}+...)$

Now I don't know what to do; usually it ends up factoring to $(1+x+x^2+...)$, but that doesn't appear to be the case here.
 [/INST] Multiplying the first two factors you find:
$x^7+ x^8+x^9+2x^{10} + $  other monomials of degree $n>10$ and, when you multiply such polynomial with the third factor, you see that the only monomial in $x^{10}$ has coefficient $2$ since all other terms have exponents $n=7,8,9$ or $n>10$.

Hint:
The OP has changed: so for this version the answer is:
Multiplying the first two factors you find:
$x^7+ x^8+x^9+3x^{10} + $  other monomials of degree $n>10$ and, when you multiply such polynomial with the third factor, you see that the only monomial in $x^{10}$ has coefficient $3$ since all other terms have exponents $n=7,8,9$ or $n>10$.
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