Circular Permutations Calculator

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.