Suppose 8 people are sitting in 8 chairs in a circle. They're playing musical chairs.
The music plays. They all get up and move.
The music ends. They all sit down.
How many ways can they sit such that nobody has the same person to their right as they did in the first orientation?
As usual for "round table" problems, the solution is invariant under rotation - i.e. the actual seat does not matter, only relative position.
I would appreciate any help on this problem. I have a feeling it would be easier to solve with inclusion-exclusion, but I can't quite define my sets correctly.