Circular Permutation Calculator

Find the number of circular permutations of n distinct items, which is (n minus 1) factorial. Also shows the linear permutation n factorial and the ratio.

Input

Find the number of circular permutations of n distinct items, equal to (n minus 1) factorial. Rotations that match are counted as one.

Number of items n

Enter the number of items to arrange in a circle as an integer of 0 or more.

Result

Circular permutations of n = 5, equal to (n minus 1) factorial

Linear permutation n factorial

120

Linear divided by circular

5 times

Digits of circular permutation

The number of circular permutations is (n minus 1) factorial, counting rotations as one. The linear permutation n factorial counts arrangements in a row and is n times larger.

How it works

A circular permutation is the number of ways to arrange n distinct items around a circle. Because rotations that look the same are counted once, the total equals the linear permutation n factorial divided by n, which is (n minus 1) factorial.
For example, seating 5 people around a round table gives (5 minus 1) factorial = 4 factorial = 24 arrangements. Rotations of the same seating count as one, so we use (n minus 1) factorial instead of n factorial.
The linear permutation n factorial counts arrangements in a row and is n times larger than the circular permutation. This tool shows the circular permutation, the linear permutation, and the ratio between them.
When n is large the result has very many digits, so beyond a certain length only the digit count is shown. All calculations use exact integer arithmetic.

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