Combinatorial proof for the identity $1 \times 1! + 2 \times 2! + ... + n \times n! = (n + 1)! - 1$

Question

What is a combinatorial proof for this identity:

$$1 \times 1! + 2 \times 2! + ... + n \times n! = (n + 1)! - 1?$$ I am trying to figure out what are both sides trying to count.

Answer 1

The $k$-th term on the left hand side counts the number of permutations of $\{1,\ldots,n+1\}$ whose last non-fixed-point is $k+1$. We choose one of the first $k$ to put in position $k+1$, then order the rest of the first $k+1$ in the first $k$ positions in any way we like.

The right hand side counts all nonidentity permutations of $\{1,\ldots,n+1\}$ (those with at least one non-fixed-point).