The problem: Prove that for $n \in \mathbb N$:
$ \left(1 + \frac{1}{n} \right)^n = 1 + \sum_{m=1}^{n} \frac{1}{m!} \left(1 - \frac{1}{n} \right) \left(1 - \frac{2}{n} \right) \cdots \left(1 - \frac{m-1}{n} \right). $
The hint is to use the binomial theorem. So the left side can become:
$ \sum_{m=0}^{n} \frac{n!}{m!(n - m)!} \left(\frac{1}{n} \right)^m $
I don't really know where to go from here, I've tried manipulating the expressions to make them look similar but I'm not really getting anywhere.