Using generating functions to find the number of solutions

Question

Using generating functions, find the number of solutions of the $u_1 +u_2..+u_6 = 23$ , $ 1≤ u_i ≤ 6,$ where $ i = 1,...6. $

Here is my working out so far: Since there are 6 integers which must sum to 23 and lie between 1 & 6, we look for the coefficient $ X^{23} $in $(X +..+X^6)^6 = X^6(1+X+...+X^6)^6$

However after this I am confused as what to do.

Answer 1

We need the coefficient of $x^{23}$ in $$(x+x^2+ \cdots +x^6)^6 = x^6(1+x+\cdots +x^5)^6 = x^6\left(\frac{1-x^6}{1-x}\right)^6$$ Hence we need to calculate the coefficient of $x^{17}$ in $$(1-6x^6+15 x^{12})(1-x)^{-6}$$ Now use the Binomial expansion for $(1-x)^{-6}$