Circular Permutation Calculator
Seating people round a table, with and without reflections.
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Results
- Circular arrangements
- 120
- Treating reflections as identical
- 60
- Linear permutations
- 720
- Seats filled
- 6
- Rotations counted as the same
- 6
Quick answer
Arranging n distinct people in a circle gives (n − 1)! arrangements because rotating everyone one seat does not create a new arrangement.
circular permutations = (n − 1)!
How to use the Circular Permutation Calculator
- 1Enter how many distinct people or objects you have.
- 2Leave the seats field blank to seat everyone, or fill it for a partial circle.
- 3Use the reflection figure for necklaces and bracelets that can be flipped over.
Frequently asked questions
Why (n − 1)! and not n!?
Fix one person as a reference point; everyone else arranges around them in (n − 1)! ways, since rotating the whole table changes nothing.
When do reflections matter?
For physical objects you can flip — a bracelet — but not for a dinner table, where left and right differ.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.