Problem 1393. A (wrong) place for everything, and everything in its (wrong) place

You have an equal number of cups and balls, each labelled from one to N. You randomly place one ball in each cup. Determine the number of possible combinations such that no balls are in the cup with a matching number. For example, if you have three balls and three cups, there are two valid solutions:

• 2, 3, 1
• 3, 1, 2

The following permutations do not meet the criteria for the reasons listed:

• 1, 2, 3 (all three balls are in the correct cups)
• 1, 3, 2 (ball 1 is in cup 1)
• 3, 2, 1 (ball 2 is in cup 2)
• 2, 1, 3 (ball 3 is in cup 3)

Good luck!