Circular Permutation Calculator
Seating people round a table, with and without reflections.
Enter your values
Enter the number of items n you are working with.
Enter the seats to fill r (optional, blank = all) you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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)!
At a glance
| What it does | Seating people round a table, with and without reflections. |
|---|---|
| Category | Math |
| Inputs needed | Number of items n, Seats to fill r (optional, blank = all) |
| Formula | circular permutations = (n − 1)! |
| Cost | Free — no sign-up, no download |
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.
Inputs explained
- Number of items n
- Enter the number of items n you are working with.
- Seats to fill r (optional, blank = all)
- Enter the seats to fill r (optional, blank = all) you are working with.
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.
What do I need to enter into the Circular Permutation Calculator?
Just 2 values: number of items n, seats to fill r (optional, blank = all). Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Circular Permutation Calculator work out the answer?
Arranging n distinct people in a circle gives (n − 1)! arrangements because rotating everyone one seat does not create a new arrangement. It applies the formula circular permutations = (n − 1)! and shows the working so you can check each step by hand.
Is the Circular Permutation Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.