The Full Question
Find the generating function and name the coefficient which would give us the solution to this problem:
count all integer solutions to $x_1 + x_2 + x_3+x_4+x_5 = 30$ where $x_i \geq 0$ and $ 0\leq i \leq 5$ and $x_2$ is even and $x_3$ must be odd.
My Work
$x_{1,4,5}$ is represented by $(1+x^1 +x^2 + x^3+\cdots + x^{27})^3$
$x_2$ is represented by $(1 + x^2 + x^4 + x^6 + \cdots + x^{28})$
$x_3$ is represented by $(x + x^3 + x^5 + \cdots + x^{27})$
We are looking for the coefficient of $x^{30}$
My Problem
The back of the book tells me I am wrong:
I don't understand this solution because say we give $x_1 = 30$ that would mean all the other $x$'s get $0$ which would mean that $x_3 = 0$ which isn't allowed because $0$ is even. So why do they include a value that would break the conditions of the question? Why is that allowed? Shouldn't we be removing illegal terms from our polynomial representation like I did?