Find the coefficient of $x^{21}$ in $(1+x^{1} + x^{2} + x^{3} + x^{4} + x^{5} + x^{6})^6$

Question

Find the coefficient of $x^{21}$ in $(1+x^{1} + x^{2} + x^{3} + x^{4} + x^{5} + x^{6})^6$

Any direction or hint would be appreciated

Answer 1

Notice that it is $$\left(\frac{x^7-1}{x-1}\right)^6$$ Now, using binomial theorem: $$(x+y)^{n}=\sum_{k=0}^n\binom{n}{k}x^{n-k}y^k$$ you get two polynomials, divide them until you reach $x^{21}$, you'll find that the solution is $7872$.