Combinatorics problem Algebra

Question

Four friends have four different balls and they decide to swap them so that everyone has a different ball. How many possible combinations are there? I know they are $9$: $2, 1, 4, 3 - 2, 3, 4, 1 - 2, 4, 1, 3 - 3, 1, 4, 2 - 3, 4, 1, 2 - 3, 4, 2, 1 - 4, 1, 2, 3 - 4, 3, 1, 2 - 4, 3, 2, 1$ but how can i calculate them without writing them down?

Answer 1

These are called derangements and have been well studied. For any $n$ there are $$ n!\Sigma_{i=0}^n\frac{(-1)^i}{i!} $$