Circular Permutations Calculator
How many distinct arrangements there are around a circle — where rotations count as the same — for n objects, or for k chosen from n, with the necklace count where reflections are also the same, and the linear count for comparison.
In a line, n things can be arranged n! ways.
How the circular permutations calculator works
In a line, n things can be arranged n! ways. Around a table, every arrangement can be rotated into n others that look the same, so there are (n − 1)! distinct seatings. If the table can also be flipped — a necklace, or a table viewed from either side — mirror images coincide too and the count halves. Choosing k of the n to seat multiplies in the ways of choosing.
Formula: circular: (n − 1)!; free (reflections equal): (n − 1)!/2; k of n in a circle: n! / (k (n − k)!)
Worked examples
| Inputs | Circular arrangements | Note |
|---|---|---|
| Eight at a round table | 5040 | 5,040 |
| Five of eight | 1344 | 1,344 |
| A bracelet of ten beads | 362880 | 181,440 free |
FAQFrequently asked questions
Why (n − 1)! and not n!?
Because rotating everyone one seat gives the same arrangement of neighbours. Fixing one person removes the rotations, and the remaining n − 1 can be arranged in (n − 1)! ways.
When do reflections count as the same?
When the object can be turned over — a necklace or bracelet — or when a seating is only about who sits next to whom, not left and right. Then each arrangement pairs with its mirror image and the count halves.
What if only some of the objects are seated?
Choose which k are seated and arrange them: n! / (n − k)! ordered selections, divided by k for the rotations. With k = n it reduces to (n − 1)!.
Why does the necklace formula need at least three?
With one or two beads a reflection is the same as a rotation, so there is nothing extra to remove. From three beads on, mirror images are genuinely different arrangements.
How big do these numbers get?
Twelve around a table is nearly forty million seatings; twenty is over 10¹⁷. The factorial grows faster than any power, which is why the page switches to exponent notation.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.