1. Find the coefficient of $x^{2}$ in the expansion of $(1+x)^{3}+(1+x)^{4}+(1+x)^{5}+\cdots+(1+x)^{n+2}$.
1. Since $(1+x)^{3}+(1+x)^{4}+(1+x)^{5}+\cdots+(1+x)^{n+2}$
$$
=\frac{(1+x)^{3}\left[(1+x)^{n}-1\right]}{(1+x)-1}=\frac{1}{x}\left[(1+x)^{n+3}-(1+x)^{3}\right]
$$

the coefficient of $x^{2}$ in the expansion is $C_{n-3}^{3}-1=\frac{(n+3)(n+2)(n+1)}{3!}-1=\frac{n^{3}+6 n^{2}+11 n}{6}$, and the coefficient of $x^{2}$ is $C_{3}^{2}+C_{4}^{2}+C_{5}^{2}+\cdots+C_{n+2}^{2}=C_{n+3}^{3}-C_{2}^{2}$