How to count circular permutations
Seating n people around a round table gives (n ā 1)! arrangements, not n!. Around a circle, shifting everyone one seat produces the same arrangement, so the n rotations of any seating are counted once. Dividing n! by n leaves (n ā 1)!. For five people that is 4! = 24.
For a necklace or bracelet, where flipping it over gives the same object, mirror-image arrangements pair up as well, so the count becomes (n ā 1)! Ć· 2. This calculator lets you pick either definition and prints which one produced the answer. If a problem says a reversed arrangement counts as the same, choose the second option.
For n of 1 or 2 a reflection changes nothing, so halving would give the wrong answer. In that case the count stays 1 and a note says so. n must be a whole number from 1 to 30.
Factorials grow so fast that from 21! onward ordinary floating point numbers can no longer hold the exact integer. This tool uses exact BigInt arithmetic, so even 30!, which has 33 digits, is exact. Values above 16 digits also appear in scientific notation, and the ratio to the straight-row count n! is shown as well.
Frequently asked questions
Around a circle, shifting everyone by one seat gives the same arrangement. The n rotations collapse into one, so n! divided by n leaves (nā1)!.
Use it when flipping the arrangement over gives the same object, as with a necklace or bracelet. Mirror images pair up, so (nā1)! is divided by 2.
With two items in a circle, a flip produces the same arrangement, so there is nothing to pair up and the count stays 1 instead of being halved.