Finding the coefficient of $x^{r}$

Question

Find the coefficient of $x^{r}$ in the expression -

$$(x+3)^{n-1} +(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^{2}+....+(x+2)^{n-1}$$

Attempt-

I tried to write it as a binomial expansion but the binomial coefficients posed a problem

Answer 1

Hint:

Apply the identity $\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\cdots+b^{n-1}\right)(a-b)=a^n-b^n$.