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.

Continuing Aravind's observation, write it like
$$
1 - \frac{k}{n} + \frac{1}{2!} \left( \frac{k}{n} \right)^2 \frac{1-1/k}{1-1/n}
- \frac{1}{3!} \left( \frac{k}{n} \right)^3 \frac{(1-1/k)(1-2/k)}{(1-1/n)(1-2/n)}
$$
$$+ \dots
+ (-1)^k \frac{1}{k!} \left( \frac{k}{n} \right)^k \frac{(1-1/k)(1-2/k)\dots (1-(k-1)/k)}{(1-1/n)(1-2/n) \dots (1-(k-1)/n)}.
$$
If $k$ and $n$ are large, this seems to be close to the $k$th degree Maclaurin polynomial
for $e^z$ evaluated at $z=-k/n$ (although I don't have an estimate of the error).
