Solution. At first this appears to be a simple permutation, where the answer is 6

At first this appears to be a simple permutation, where the answer is 6! However, if we look at the two situations below, where the chairs are labelled from 1 to 6 and the people from A to F, we can see that they appear as different permutations, but are actually the same.

Because of the actual situation, every person has the same people on either side in both cases. If the chairs had been distinguishable, then this would no longer be the case.

Hence for every permutation of people sat around the table, there are five more permutations which are the same and hence the answer is six times too big. These are shown in the diagram below.

Therefore the number of ways that six people can be sat around a circular dining table is

NOTE! For any situation like this the answer can be generalised to