Is
$$\sum_{k=0}^n \binom{n}{k}(-1)^k\frac{(n-k)^n}{n!} = 1$$
a common identity?
After extensive googling, I am unable to find anything on it specifically.
It came up as part of an answer to a previous binomial question of mine: Binomial proof of $n! = n^r - n(n - 1)^r + \frac{n(n-1)}{2!}\left(n -2 \right)^r - \dots$ when $r = n$. I follow the entire argument of the accepted answer, but I don't see why $\sum_{k=0}^n \binom{n}{k}(-1)^k\frac{(n-k)^n}{n!} = 1$. I'd prefer clarification within the context of the linked question, but combinatorial proofs are nice too.
Also, based on equating coefficients, the answer in the linked question implies that $$ \left(1 + \frac{x}{2!} + \frac{x^2}{3!} + \cdots \right)^n = 1$$
as well. I'm unfamiliar with this identity too, although it sort of looks like a Taylor series for...$1$??
Thank you for any assistance.