The Round Table

Question

In how many distinct ways can 8 people be seated around a circular table, where two seatings are the same if one is a rotation of the other?

Accepted Answer

Fix person A in a seat, every circular arrangement has exactly one rotation with A in that seat, so this counts each arrangement once. The remaining 7 people fill 7 seats in order: 7! = 5040. Generally: (n-1)! circular arrangements of n people. If reflections (clockwise vs counterclockwise) are also considered identical, e.g. a key ring, divide by 2: frac(n-1)!2. The principle: count with symmetry by fixing a representative, or count everything and divide by the symmetry group's size (n! linear seatings div n rotations = (n-1)!). State which symmetries you're quotienting by before computing, that's what the interviewer is checking.

The Round Table

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