Study the concept of **Derangements** from a good resource.

Derangement is basically the opposite of arrangement. In simple words, given $n$ objects and $n$ locations, how many ways can you arrange them such that all $n$ are in the wrong spots.

There is an accurate formula for this, but it can be approximated with $\frac{n!}{e}$

For $n = 5$, derangements $= \frac{5!}{e} \approx 44$

So there are $44$ ways to put all $5$ letters into the wrong envelopes, and $5!$ total ways of arranging the $5$ letters

$\therefore$ probability that __all__ $5$ letters are in wrong envelopes $= \frac{44}{5!} = \frac{44}{120}$

$\therefore$ probability that __at least one__ letter is in the right envelope $= 1 -\frac{44}{120} = \frac{76}{120} = \frac{19}{30} $