The six members of the yearbook staff sit around the circular table in their office. How many different seating arrangements are there of this group of people?

Answer 1

imagine the members in a row,
they can be arranged in 6! or 720 ways

when seated around a table everybody could shift one seat to the left without changing the order.
This can be done 6 times, so the 720 includes 6 duplications of each seating.

So the number of ways to seat them around a circular table is 720/6 or 120 ways.

Note: This argument assumes that the seats are indistinguishable, otherwise the number would be 6!
Have you come across the questions "In how many ways can 6 keys be arranged around a key-ring" ?

Answer 2

It helps to start with a simpler problem. Try solving it manually, by drawing out each possible seating arrangement, with three members instead of six. The problem specifies a circular table, so we can probably assume that there is a difference between circular and linear arrangements. What I did is write out 123, 132, 213, 231, 312, 321, which are all the possible linear arrangements. Now to make them into circular arrangements, where there is no beginning and end, 123 becomes ...12312312... which also contains 231 and 312. The other arrangement turns out to be ...13213213... (I'm assuming it doesn't matter which chair each person sits in, only their positions relative to each other.) Three factorial = 3! = 3x2x1 = 6 linear arrangements and 2! = 2 circular arrangements. This pattern becomes clear if you do the problem again with four people. The answer to the original problem is 5! = 120