Combinatorical meaning of an identity involving factorials While solving (successfully!) problem 24 in projectEuler I was doodling around and discoverd the foloowing identity:
$$1+2\times2!+3\times3!+\dots N\times N!=\sum_{k=1}^{k=N} k\times k!=(N+1)!-1$$
While this is very easy to prove, I couldn't find a nice and simple combinatorical way to interpret this identity*. Any ideas?

*That is, I do have a combinatorical interpretation - that's how I got to this identity - but it's not as simple as I'd like.

The number of ways you can sort a set of consecutive numbers starting from $1$ and none of which is larger than $N$ and then paint one of them blue is $(N+1)!-1$.
