Find coefficient in this expression

Question

I want to find the coefficient of $x^{21}$ in this expression: $(x^3+x^4+x^5+\ldots +x^{10})^4$. The first thing I did was $(x^3+x^4+x^5+\ldots +x^{10})^4=(x^3)^4(1+x+x^2+\ldots+x^7)^4$. So the problem is reduced to finding the coefficient of $x^9$ in the expression $(1+x+x^2+\ldots+x^7)^4$. However, I am not able to find that coefficient :/

Answer 1

$\mathbf{\text{HINT}}:$Remember that $$(1+x+x^2 +...+x^7)^4= \bigg(\frac{1-x^8}{1-x}\bigg)^4$$ , so you have binomial expressions.

So, $(1-x^8)^4 (1-x)^{-4}$ , then $(1+(-x^8))^4 (1-x)^{-4}$ ,

you will just find coefficients of binomial expressions such that

For $(1+(-x^8))$ : $\binom{4}{m}(-x^8)^m$

For $(\frac{1}{1-x})^4$ : $\binom{n+4-1}{n}x^n$ , where $8m+n=9$

Find all $m,n$ values satisfying the equation such as $(0,9),(1,1)$ , then put them into formulas.

Then , $$1 \times\bigg( \binom{9+4-1}{9}\bigg)+ (-4)\times \bigg(\binom{1+4-1}{1}\bigg)=204$$