Given a finite set $S$ with $|S|= n$, what is the number of permutations $\sigma$ such that
$$\sigma(i) \not= i, \forall i \in S $$
That is, permutations where every element is interchanged with another element. (The permutations which can be described as one cycle containing exactly $n$ elements?)
My naive failed attempt at a solution:
$|S|= n$, consider the first element. We have $n-1$ places to put it. Considering the second element the number of choices depends on where we put the first element. I reckon it's either $n-1$ or $n-2$. And the problem continues since the next choice depends on the previous one. I'm Lost.