The coefficient of  $x^r(0\le r\le (n-1))$ in the expansion of ${(x+3)}^{n-1}+{(x+3)}^{n-2}(x+2)+{(x+3)}^{n-3}{(x+2)}^2+\cdots +{(x+2)}^{n-1}$ is 

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

we know that

$a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots +b^{n-1})$

Given expression

$\frac{{(x+3)}^n-{(x+2)}^n}{(x+3)-(x+2)}=\frac{{(x+3)}^n-{(x+2)}^n}{1}$

Coefficient of $x^r$ in  ${(x+3)}^n-{(x+2)}^n$

$\begin{array}{l}{}^n{\text{C}}_{n-r}{3}^{n-r}-{}^n{\text{C}}_{n-r}{2}^{n-r}\\ ={}^n{\text{C}}_r(3^{n-r}-2^{n-r})\end{array}$