Find coefficient of x in a generating function

Question

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.

Answer 1

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 $n>10$.

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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 $n>10$.

Answer 2

Hint : coefficient of $x^3$ in $(1+x^2+x^3)(1+x+x^4)(1+x^5+x^{10}+x^{15}+...)$ is $2$, because $x^3=x^3\times 1 \times 1, x^3= x^2\times x \times 1$

Answer 3

From the sum $\frac{1}{1-x^5}$ you only need to consider the first two terms - other will give you a higher power. $10 = x_1 +x_2+ x_3$ (with terms all three sums). Can you idenify $x_k$?