I have often heard a nice way that $e$ can appear in probability theory -- that, say, I had an abitrarily large box of specialty chocolates (so they are all unique), and I mixed them all up. The probability that such a mixture of chocolate is completely mixed (as in there is no chocolates that are in their original place) approaches $\frac{1}{e}$
ATTEMPT:
Let $D_n$ be the number ways of dearranging. So we might realise that for some pair of chocolates $A,B$ there is the case where $A$ gets swapped with $B$ and thus the remaining number of ways to dearrange becomes $D_{n-2}$, Or the second case may be when $A$ gets placed into the place for some $C\neq B,A$. And then $B$ is then put into one of the remaining $(n-2)$ positions (as it cannot occupy its oringal slot, or the slot that $A$ has just been put into.
At this point, I feel that their is some obvious recurrence formula (as the $D_{n-2}$ term appears, and this would appear to continue) but I can't quite see it. Some help would be appreciated!