Seating eight guests at a circular table

Question

Eight guests are seated around a circular dinner table with no designated head. Seatings related by a rotation are considered identical. How many distinct seatings are there?

Accepted Answer

Fix one guest's seat to anchor the circle and eliminate rotational duplicates. The remaining 7 guests then arrange themselves in the 7 seats around the table: 7! = 5040. Equivalently, \dfrac{8!}{8} = \dfrac{40320}{8} = 5040 = (8-1)!. Takeaway. The round-table rule scales directly: n guests give (n-1)! seatings. Eight guests give 7! = 5040.

Seating eight guests at a circular table

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