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Circular Permutations Calculator

Count round-table arrangements.

How it is calculated

For n distinct people or objects placed around a circle, where turning the whole circle does not create a new arrangement, the count is (n − 1)! = (n − 1) × (n − 2) × … × 2 × 1. One item is fixed as a reference seat, and the other n − 1 are arranged in order relative to it. The tool multiplies the integers from 2 up to n − 1 and prints the exact whole number with thousands separators. The input must be a whole number from 1 to 20; 0, negatives, decimals such as 7.5, and anything above 20 give no result. For n = 1 and n = 2 the answer is 1, since a lone person has one seat and two people can only sit opposite each other. The count treats a mirror image as a different arrangement: sitting clockwise as A, B, C is not the same as A, C, B.

Worked example

Eight guests are seated at a round table. The linear count would be 8! = 40,320, but rotations of the same seating are identical, so we fix one guest and arrange the other seven. That gives (8 − 1)! = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 distinct seating plans. Dividing 40,320 by 8 gives the same number, since each circular seating appears in 8 rotations.

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Common mistakes

  • Using n! instead of (n − 1)!. For 8 guests that is 40,320 rather than 5,040, because it counts every rotation of the same seating as new.
  • Applying it to identical or repeated items. The formula needs every person or object to be different; with repeats, you would divide by the factorials of the repeated counts.
  • Counting a necklace or bracelet with it. A bracelet can be flipped over, so mirror images are the same and the count is about half of this result; this tool does not halve it.
  • Leaving out a person who has a reserved seat. Count everyone, host included: with 8 people the answer is 7! = 5,040, and fixing the host's seat does not change that number.
  • Entering a number above 20 or a fraction such as 7.5. The tool needs a whole number from 1 to 20 and shows no result outside that.

About this calculator

A circular permutation counts the distinct ways to arrange n distinct objects around a circle, where arrangements that differ only by rotation are considered the same. While n objects in a straight line can be ordered n! ways, fixing one object as a reference point to cancel out the rotational duplicates leaves only (n−1)! genuinely different circular arrangements — for six people around a round table, that is 5! = 120 seatings rather than 720.

This distinction matters wherever position is relative rather than absolute: event planners use it to count round-table seating arrangements, puzzle and combinatorics courses use it to illustrate why “linear” and “circular” counting problems diverge, and it shows up in scheduling problems like round-robin tournament rotations where only who sits next to whom matters, not an arbitrary starting seat.

This calculator applies the (n−1)! formula directly to your value of n, giving you the exact count of unique circular arrangements — and it’s worth noting this differs from “necklace” counting, which further divides by 2 when flipping (reflection) is also considered equivalent.

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