# 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:

$$20+2-1 \choose 20$$ $$-$$ $$7 \choose 1$$ $$15+2-1 \choose 15$$+$$7 \choose 210+2-1 \choose 10$$-$$7 \choose 345+2-1 \choose 5$$+$$7 \choose 40+2-1 \choose 0$$

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?

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