Combinatorics with surprising results

Given n letters and n envelopes, in how many ways can the letters be placed in the envelopes, so that no letter is in the correct envelope.
You will find euler's number in the solution.

I thought Euler's number only turned up in the limiting ratio?

How? Can you show me that? Wouldn't it be like n! - 1 or something?

How many permutations of Sn are there that fix no element {1,...,n}?

Using the inclusion-exclusion principle, the number of permutations that fix elements is equal to:
(nC1)*(n-1)! - (nC2)*(n-2)!.... (-1)^n +(nCn)*0!

Then, the number that don't fix elements is n!-that = (nC2)*(n-2)!-.... (-1)^n+1 +(nCn)*0!
= n!/2! - n!/3! ...(-1)^n+1 n!/n!
=n!(1/2!..... (-1)^n+1 1/n!)
In the limit n~inf, this is ~ to n!/e with some further proof.

Oops, when I say further proof I mean, barely any at all. Just divide by n!/e.