<s>[INST] Find number of positive integer solutions to the equation $a+b+c=15$ where a,b are even and c is odd This is equivalent to:
Coefficient of $x^{15}$ in $(1+x^2+x^4+\cdots+x^{14})^2(x+x^3+\cdots+x^{15})$
Or
Coefficient of $x^{14}$ in $(1+x^2+x^4+\cdots+x^{14})^3$
Is this approach correct?
 [/INST] Why not just solving the problem with a direct approach? You know that $a$ and $b$ are even and $c$ is odd, so $a = 2A, b = 2B$, and $c = 2C+1$, with $A,B$ positive integers and $C \geq 0$. Now the problem amounts to solving
$$2(A+B+C)+1 = 15,$$
so you want $A,B > 0$ and $C \geq 0$ integers such that
$$A+B+C = 7.$$
You can just do this by trying case by case.
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