How many ways can 7 people be seated at a round table? In how many ways can 7 people be seated at a round table if they can sit anywhere? The answer is 6!=720 I expected 7! I don't understand why. Does it have anything to do with the fact that the table is round?

Imagine 7 people sat around a table, then ask everyone to move to the seat to their right, and to do this again and again and again - all of these arrangements would be the same, it's essentially just the circle which has been rotated, there is no difference in the arrangement! These people could keep doing this, moving to the seat to their right until they're back to the 'seat' they started at, this would happen 7 times, so we divide by this 7 as these are not unique combinations.

In general, for n-seats there are (n-1)! ways