Help with finding the coefficient in a generating function expansion

by LoudAnnoyance   Last Updated May 16, 2019 06:20 AM

Find the coefficient of ${x^{20}}$ in the expansion of the generating function g(x) = $\frac{5{(1-x^5)^7}}{(1-x)^{2}}$

I broke the function into two components: $5{(1-x^5)^7}$ and $\frac{1}{(1-x)^2}$

Because I'm looking for the ${x^{20}}$ coefficient, I have 5 terms:

$a_0*b_{20}$ + $a_5*b_{15}$ + $a_{10}*b_{10}$ + $a_{15}*b_5$ + $a_{20}*b_0$

which gives me:

$20+2-1 \choose 20$ $-$ $7 \choose 1$ $15+2-1 \choose 15$$ $+$ $$7 \choose 2$$10+2-1 \choose 10$$ $-$ $$7 \choose 3$$45+2-1 \choose 5$$ $+$ $$7 \choose 4$$0+2-1 \choose 0 $

I believe that I multiply the $5$ in the first polynomial to the coefficient I find, but my answer comes out to be $-35$ which isn't possible. Any suggestions on how to fix this issue?

Tags : combinatorics


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