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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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