Evaluation

Note that $\frac{k \times k! \times \binom{n}{k}}{n^k}\; = \; \frac{k! \binom{n}{k}}{n^{k-1}} - \frac{(k+1)!\binom{n}{k+1}}{n^k} \hspace{2cm} 1 \le k \le n-1$ and hence $\begin{aligned} \sum_{k=1}^n \frac{k \times k! \times \binom{n}{k}}{n^k} & = \; \sum_{k=1}^{n-1}\left[ \frac{k! \binom{n}{k}}{n^{k-1}} - \frac{(k+1)!\binom{n}{k+1}}{n^k} \right] + \frac{n!}{n^{n-1}} \\ & = \; \left[\binom{n}{1} - \frac{n!}{n^{n-1}}\right] + \frac{n!}{n^{n-1}} \; = \; \boxed{n} \end{aligned}$