> Score Probability

> 0 0.3679

> 1 0.3679

> 2 0.1839

> 3 0.0613

> 4 0.0153

> 5 0.0031

For 0, it's a well known [1] result 1/e, I remember a puzzle where people left their hat and then pick one randomly.

Looking at the table it looks like the general formula is 1/(e*n!) that is a Possion distribution. Compare with https://en.wikipedia.org/wiki/Poisson_distribution#Examples_...

Anyway, I'm not sure if my observation helps too much to solve the original problem.

[1] at least I know it :)