Circular Permutation Calculator

Enter Calculation

Number of objects (n)

Results

Number of circular permutations

24

distinct circular arrangements

Number of objects (n)	5
Formula	(n − 1)!
Computation	(5 − 1)! = 24

What is a circular permutation?

A circular permutation counts the number of distinct ways to arrange n distinct objects around a circle when rotations of the same arrangement are considered identical. Unlike a row (a linear arrangement), a circle has no fixed starting point, so rotating everyone one seat to the left produces the same arrangement. This calculator returns the exact count, \((n - 1)!\), using arbitrary-precision arithmetic so even large values stay exact.

Circular arrangements that differ only by rotation are counted as the same.

How to use this calculator

Enter the number of distinct objects n (a positive integer, n ≥ 1) and read off the number of circular permutations. For example, the number of ways to seat n people around a round table, or arrange n distinct beads in a fixed ring orientation, is \((n - 1)!\).

The formula explained

The number of linear permutations of n distinct objects is n!. In a circle, each unique arrangement can be rotated into n different but equivalent positions (one for each possible starting object). Dividing the linear count by these n rotations gives:

$$\frac{n!}{n} = (n - 1)!$$

Note: this is the standard circular permutation. It does not treat mirror reflections (clockwise vs counter-clockwise) as the same. If reflections were also identical — as with a flippable necklace or bracelet — the count would be \((n - 1)! / 2\) for n ≥ 3.

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Fixing one object's seat removes rotational duplicates, giving (n-1)! arrangements.

Worked example

For n = 5: $$(5 - 1)! = 4! = 4 \times 3 \times 2 \times 1 = 24.$$ So five distinct people can be seated around a round table in 24 distinct ways once rotations are treated as the same.

FAQ

What is the answer for n = 1 or n = 2? For n = 1, \((1 - 1)! = 0! = 1\). For n = 2, \((2 - 1)! = 1! = 1\) — two objects in a circle have only one distinct arrangement up to rotation.

Why divide by n instead of subtracting? Because each circular arrangement corresponds to exactly n equivalent linear orderings (one per rotation), you divide the total n! by n, which simplifies to \((n - 1)!\).

Does this count necklaces or bracelets? No. This computes the standard circular permutation \((n - 1)!\). Necklace/bracelet counts that also collapse mirror images use \((n - 1)! / 2\) for n ≥ 3.