Find the coefficient of $x^n$ in $(x^2 +x^3 +x^4 +\cdots)^5$
As you had before, we have $$f(x)=(x^2+x^3+\ldots)^5$$ $$f(x)=x^{10}(1+x+\ldots)^5$$ $$f(x)=\frac{x^{10}}{(1-x)^5}$$ $$f(x)=\frac{x^{10}}{(1-x)^5}$$ $$f(x)=x^{10}\sum_{k=0}^\infty \binom{k+4}{4}x^k$$ $$f(x)=\sum_{k=0}^\infty \binom{k+4}{4}x^{k+10}$$ Hence, the coefficient of $x^n$ is $\binom{n-6}{4}$