How to Calculate Circular Permutations

Calculates the number of ways to seat n people in a circle, (n − 1)!. Arrangements that differ only by rotation count as one, which is why it is (n − 1)! and not n!.

Explanation

A circular permutation counts the ways of arranging $n$ items around a circle.

(n-1)!

Arranged in a row there would be $n!$ orders, but on a circle any arrangement that is a rotation of another counts once, so we divide by $n$ and are left with $(n-1)!$ .

Example

Seat 5 people at a round table.

(5-1)! = 4! = 4 \times 3 \times 2 \times 1 = 24

In a row there would be $5! = 120$ orders, but shifting everyone one seat round the table changes nothing, so the 120 fall into groups of 5 identical arrangements: $120 \div 5 = 24$ .

The trick

Fix one person in place. Once someone is pinned to a reference seat, the remaining $n-1$ people simply have to be arranged in a row, which immediately gives $(n-1)!$ .

Watch out

If flipping the arrangement over also counts as the same thing, divide by 2 as well. Threading $n$ beads onto a necklace gives $\dfrac{(n-1)!}{2}$ arrangements for $n \ge 3$ . A dinner table cannot be flipped, so no extra division there
If the seats are numbered, so that "seat A" and "seat B" are distinguishable, it is not a circular permutation at all but an ordinary one, with $n!$ arrangements