Find coefficient of $x^{10}$ in $\left(1−x^7\right) \left(1−x^8\right) \left(1−x^9\right) (1−x)^{-3}$

Question

find the coefficient of $x^{10}$ in this expansion: $$ \left(1−x^7\right) \left(1−x^8\right) \left(1−x^9\right) (1−x)^{-3} $$ Please help me solve this question

Answer 1

HINT

Expand $(1-x)^{-3}$ into a series, writing $$ (1-x)^{-3} = \sum_{k=0}^\infty a_k x^k $$ so you would get $$ \left(1−x^7\right) \left(1−x^8\right) \left(1−x^9\right) \sum_{k=0}^\infty a_k x^k $$ and think about how you can get a product of $x^{10}$ from these 4 factors:

• if the first factor is present, you get $-x^7 a_3 x^3$
• if the second factor is present, you get $-x^8 a_2 x^2$
• if the third factor is present, ...
• if on the fourth factor is present, ...

and the result is the final sum of those.

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HINT 2

Alternative way suggested in the comments below: note that $$ \frac{1}{1-x} = 1 + x + x^2 \ldots $$ so you can write $$ \frac{1-x^7}{1-x} = \frac{1}{1-x} - x^7 \frac{1}{1-x} = 1 + x + x^2 \ldots - x^7 - x^8 \ldots = 1 + x + \ldots x^6 $$ and your products become $$ (1+x+\ldots x^6)(1+x+\ldots x^7)(1+x+\ldots x^8) $$