Circular Permutations Calculator
Count circular arrangements of n distinct objects around a fixed circle using (n-1)! formula.
Compare with linear permutations and necklace arrangements.
Circular Permutations
A circular permutation is an arrangement of objects in a circle, where only relative order matters — not absolute position. Because a circle has no fixed starting point, rotations of the same arrangement are considered identical.
Formula
For n distinct objects arranged in a circle:
Circular permutations = (n − 1)!
We fix one object in place to eliminate rotational duplicates, then arrange the remaining (n − 1) objects in the remaining positions: (n − 1)!
Comparison
| Type | Formula | Example (n=4) |
|---|---|---|
| Linear permutations | n! | 4! = 24 |
| Circular permutations | (n − 1)! | 3! = 6 |
| Necklace (flip = same) | (n − 1)! / 2 | 3 |
A necklace arrangement also treats flipped (mirror image) arrangements as identical. This is relevant for physical objects like keychains, bangles, or beads that can be flipped over.
Examples
- 3 people at a round table: (3-1)! = 2 ways
- 5 people at a round table: (5-1)! = 24 ways
- 6 keys on a ring (necklace): (6-1)! / 2 = 60 ways
Why Does This Matter?
Linear vs circular is a classic combinatorics distinction. Seating people at a round table, arranging objects in a ring, or scheduling tasks in a cycle all require circular permutation formulas.
Factorial Growth
Circular permutations grow very rapidly with n. For n = 10, there are 9! = 362,880 distinct circular arrangements.