How many ways are there to seat six people around a circular table

How many ways are there to seat six people around a circular table where two seatings are considered the same when everyone has the same two neighbors without regard to whether they are right or left neighbors?

I know we have to apply division rule. I also know by rotating at six position we overcount by factor of 6 but that gives me answer of 120, while the correct answer is 60.

How to solve this?

• $6$ people sit on a circular table in $(6-1)! = 120$ ways Oct 23, 2020 at 8:24
• Now if left or right neighbours do not matter, that is $1/2$. Oct 23, 2020 at 8:25
• Said differently: For each of your solutions, flip one clockwise-to-counterclockwise. They're equivalent. Hence divide by $2$. Oct 23, 2020 at 19:19

All the guests can sit around the table in $$6! = 720$$ ways. Since two seatings are considered equal if they all have the same neighbor, the location of the first person, does not matter. Hence, a division by $$6$$. Also, since the left/right orientation is not an issue, we can 'flip' (place the first person at a seat, and instead of adding persons to the right, do it to the left) the seating and thus divide by $$2$$. This gives $$60$$ seatings in total.