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

I came across the built-in Subfactorial function in the documentation and altered one of the examples to produce:

Remove[teleSecretSanta];
teleSecretSanta[dims_Integer] :=
 With[{spec = Range[dims]},
  With[{
    perms = Permutations[spec],
    casesToDelete = DiagonalMatrix[spec] /. {0 -> _}},
   DeleteCases[perms, Alternatives @@ casesToDelete]
   ]
  ]

One can use Subfactorial to check the function.

Length[teleSecretSanta[4]] == Subfactorial[4]

As in Mr.Wizard's answer, I suspect teleSecretSanta can be optimized via SparseArray. However, I'm too drunk at the moment to attempt such shenanigans. (kidding... I'm actually too lazy and stupid.)