Circular Permutations Calculator
Count the ways to seat n distinct people around a circular table — (n − 1)! for rotations, or (n − 1)!/2 when reflections are the same.
How it works
(n − 1)! for rotations · (n − 1)! / 2 when reflections are identical (n ≥ 3)
Around a circle there is no fixed “first” seat, so the n rotations of any seating are all the same arrangement — dividing the n! linear orderings by n leaves (n − 1)!. If the arrangement can also be flipped over (a bead necklace, say), each pattern and its mirror image coincide, halving the count again for n ≥ 3. Fixing one person’s seat and arranging the rest is the intuitive way to see the (n − 1)! result.
Worked example
Seating 6 people around a round table: (6 − 1)! = 5! = 120 distinct arrangements. If mirror images count as the same, that halves to 60.
Sources: Cyclic permutation — Wikipedia
Frequently asked questions
Why (n − 1)! instead of n!?
A circle has no starting point, so rotating everyone by one seat is not a new arrangement. The n rotations collapse to one, dividing n! by n to give (n − 1)!.
When do I divide by 2 as well?
When the arrangement can be flipped over and still count as the same — like beads on a necklace or a keyring. Then each pattern equals its mirror image, halving the count for n ≥ 3.
Does a round table with numbered seats use this?
No. If the seats are labelled or fixed, position matters and there are n! arrangements, just like a straight line.
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