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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:

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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?

Dunka
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  • There seems to be no reason to leave out $x^{29}$ in the third term. In the second term, the powers of $x$ are even, where did the $x^{27}$ come from? – André Nicolas Feb 01 '15 at 03:48
  • @AndréNicolas That was a typo on the second term, sorry about that, just fixed it now. I figured if I included $x^{29}$ then in that case we might end up computing a case where $x_2 = 1$ – Dunka Feb 01 '15 at 03:58
  • You have mentioned that 0 is even and why should you not include 1 in the second term for x2 – Satish Ramanathan Feb 01 '15 at 04:00
  • @satishramanathan Because of the restriction where $x_2$ must be even – Dunka Feb 01 '15 at 04:08
  • I realize why my solution doesn't work. Thanks guys. – Dunka Feb 01 '15 at 05:51

0 Answers0