Circular Permutation Calculator
Find how many unique ways there are to arrange people around a table or beads on a string. Rotational symmetry is accounted for automatically.
What is a circular permutation?
A circular permutation is an arrangement of items in a circle. Because rotations are considered identical, the total number of unique circular arrangements is (n - 1)! instead of n!.
How do you calculate circular permutations?
For n distinct items, the formula is (n - 1) factorial. For example, if 5 people are sitting around a round table, there are (5 - 1)! = 4! = 24 unique seating arrangements.
What if reflections are also identical (like beads on a necklace)?
If the arrangement can be flipped (like a necklace or a keychain), the formula becomes (n - 1)! / 2. This is because every circular arrangement and its mirror image are considered the same.
Why do we subtract 1 from n?
In a circle, there is no fixed starting position. We "fix" one person in a specific spot to act as a reference point, and then arrange the remaining (n - 1) people around them.