Secret Santa - Generating 'valid' permutations

Question

My friends invited me home to play the game of Secret Santa, where we are supposed to draw a lot & play the role of \'Santa\' for a friend in the group.

So, we write

Answer 1

A permutation that maps no element to itself is a derangement. As n increases, the fraction of derangements approaches the constant 1/e. As such, it takes (on average) e tries to get a derangement, if picking a permutation at random.

The wikipedia article includes expressions for calculating explicit values for small n.