Probability that a random permutation has no fixed point among the first $k$ elements Is it true that $\frac1{n!} \int_0^\infty x^{n-k} (x-1)^k e^{-x}\,dx \approx e^{-k/n}$ when $k$ and $n$ are large integers with $k \le n$?
This quantity is the probability that a random permutation of $n$ elements does not fix any of the first $k$ elements.

This is not an answer but an observation that the quantity is equal to $\sum_{i=0}^{k} \frac{{(-1)}^i}{n!}{k \choose i}(n-i)!$.
