What is the expectation of the following random variable

$S_n=X_1+...+X_n \\ P(S_n \leq t),t<n=\text{The volume between the axises and the hyperplane X_1+X_2+...+X_n=t which is } \frac{t^n}{n!} \\ P(S_n \leq t)=\frac{t^n}{n!},t<n\\ f_{S_n}(t)=\frac{t^{n-1}}{(n-1)!},t<n \\ P(N=n)=\int\limits_0^1f_{S_{n-1}}(t)P(X_n>1-t)dt=\int\limits_0^1\frac{t^{n-2}}{(n-2)!}t dt=\frac{1}{n(n-2)!},n\geq2$

Supplement to the allready accepted answer of hhsaffar:

$P\left[N>n\right]=P\left[S_{n}\leq1\right]=\frac{1}{n!}$ and $E\left[N\right]=\sum_{n=1}^{\infty}P\left\{ N\geq n\right\} =\sum_{n=0}^{\infty}P\left\{ N>n\right\} =\sum_{n=0}^{\infty}\frac{1}{n!}=e$