Question

In how many ways can 6 people be seated around a table?

Answer 1

If the seats are distinguishable, 720. If indistinguishable, 120.

Explanation:

If we define "sitting around a table" to mean that each seat is distinguishable (for instance, each chair is a different colour and in a fixed place at the table, or the table itself is arranged in such a way that we can describe each seat differently, such as Right Side Facing the Window, Left Side Facing the Window, Right End, Left End, Right Side Facing Away from the Window, Left Side Facing Away from the Window), then there are `6! = 720` ways to arrange six girls around the table.

However, if we define "sitting around a table" to mean that each seat is indistinguishable, and so therefore we are more concerned with each girl having a different person sit next to them, then we need to divide the number of ways we can seat them by 6 - this will eliminate the double counting (girls 1, 2, 3, 4, 5, 6 in seats A, B, C, D, E, F; and then in seats B, C, D, E, F, A; and etc), which gives us:

`(6!)/6=5! = 120`