I just started learning about generating functions and was trying to solve a problem on my own. If anyone can give me feedback on my solution. Letting me know if it is right or wrong or any place I made a mistake it will be greatly appreciated! Thank you! 
According to the textbook the solution is C(12+5-1,12)-5 X C(7+5-1,7)+C(5,2)X C(2+5-1,2) but when I evaluate that and compare it to my solution I get a different number.
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Hint: Note that $$\frac{(1-x^5)^5}{(1-x)^5}=(1-x^5)^5 \cdot (1-x)^{-5}$$ and then use the Binomial Theorem for a negative power to expand $$(1-x)^{-5} = \sum_{i=0}^{\infty} \binom{5+i-1}{i} x^i$$
Your mistake is in expanding $(1-x)^5$ instead of $(1-x)^{-5}$.