Suppose I have a set $S$ with $n$ items, and I create the ordering (e.g. $\{a, b, c, d\}$) using every element of S exactly one. How many orderings could I create with these same items such that:
-All elements of S are in the new ordering
-Each element occurs only once.
-Element $i$ of the first ordering does not match element $i$ of the second ordering
(e.g. so if the first element is a in the first ordering, so the first element in the second ordering can't be a).
I think maybe a recursive/inductive approach could be useful, but I suspect there might be a method that doesn't require this and instead simplifies with combinatorics or something.
