You have to treat this a limit.
If there are $n$ postmen delivering one letter each, and $m$ mailboxes, so if the postmen deliver at random then the probability one postman puts the letter in the wrong mail box is $\left(1-\frac{1}{m}\right)$, and so (assuming independence) the probability all the letters go in the wrong mail boxes is $\left(1-\frac{1}{m}\right)^n.$
You want the limit of this as $n \to \infty$ and $m \to \infty$. Sadly that limit is not well defined.
If $n = m $ then you can use $\lim_{n \to \infty} \left(1-\frac{1}{n}\right)^n = e^{-1} \approx 0.367879$ and this may be what your friend was thinking.
But if $\lim n/m = k$ then $\lim_{n \to \infty} \left(1-\frac{1}{m}\right)^n = e^{-k}$ and this can take any value between $0$ and $1$, achieving the extremes if $k$ is infinite or zero.