All n items: n!. Choosing k of n in order: n!/(n-k)!.
Circular arrangements divide by n, because rotations are equivalent: (n-1)! ways to seat n people at a round table. This is a standard interview adjustment and it is easy to forget.
Repeated items divide by the factorial of each repeat count. The letters of MISSISSIPPI arrange in 11!/(4! 4! 2!) ways.
Derangements - permutations with no fixed point - number approximately n!/e, so the probability that nobody gets their own hat back converges to about 36.8%, and does so very quickly.