TITLE: How many ways can seven people sit around a circular table?
QUESTION [34 upvotes]: How many ways seven people can sit around a circular table?
For first, I thought it was $7!$ (the number of ways of sitting in seven chairs), but the answer is $(7-1)!$.
I don't understand how sitting around a circular table and sitting in seven chairs are different. Could somebody explain it please?

REPLY [3 votes]: Number the positions around the circle from $1$ to $n$. Cut the circle at any position, say $1$, and lay out the circle as a straight line. The permutations of the $n$ points are $n!$ in number. But for a circle you could have cut the string at any point and still obtained a permutation of $n!$. Therefore we divide by $n$ to get $(n-1)! $