Seating six people around a round table

Question

Six people sit around a round table. Two seatings count as the same if one is just a rotation of the other (there are no numbered seats and no fixed head of the table, so what matters is only who sits next to whom).

How many distinct seatings are there?

Accepted Answer

If the seats were fixed and labeled, there would be 6! = 720 ways. But around a plain round table each seating can be rotated into 6 positions that all look the same, so we have counted every genuine arrangement 6 times. Divide: \frac{6!}{6} = \frac{720}{6} = 120 = (6-1)! = 5!. A cleaner way to see it: seat one person first to "anchor" the circle and kill the rotational symmetry, then arrange the remaining 5 people in the 5 seats around them: 5! = 120. Takeaway. Arranging n distinct people around a circle (rotations equivalent) gives (n-1)!. Six people give 5! = 120.

Seating six people around a round table

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