A group of 10 people are seated at a circular table. They hadn’t noticed that they had places designated for them, and none of them sat in the right place. If the table is rotated by one position, calculate the probability that 0,1,2,3,4,5,6,7,8,9,10 of the people are then seated at their correct position. Similarly, calculate the probabilities if the table is rotated (in the same direction) one more position, etc.
This problem was inspired by Ragknot. I haven’t thought about it much, my initial belief is that it’s fairly easy, but I could be wrong. If I’m right about it being fairly easy, then it should also be the case that you can give a general solution, rather than loads of numbers.
For consistency in the solutions, consider that the names of the people are “1″,”2″,…”10″, and that the seats are labelled in the order 1,2,3,…,10 (like a clock that only has 10 hours on it’s dial). The clock/table is only rotated (one place) anti-clockwise, so that someone who started in seat n will be in seat n+1 (mod 10).
In case you don’t know where to start, this problem is all about derangements