Permutations and circular arrangements

Content updated: October 22, 2026

Permutations count arrangements of all the distinct elements. For arrangements around a circle, the count depends on whether positions are distinguished or arrangements differing only by rotation are identified.

On this page: The idea · Method · Examples · Practice · Further learning

1. Permutations of distinct elements

Let $A$ be a set of $n$ distinct elements, with $n\ge1$. A permutation is an ordered list containing each element of $A$ exactly once. Write $P_n$ for the number of permutations.

There are $n$ choices for the first element, $n-1$ for the second, and so on until the last position has $1$ choice. By the multiplication principle,

Each list uniquely determines the choices by position, so no arrangement is repeated or omitted. For the empty set, there is exactly one empty list; the convention $0!=1$ corresponds to this case. If some objects are regarded as identical, the distinctness hypothesis fails and $n!$ cannot be applied directly.

2. When do circular permutations apply?

For circular permutations, arrangements differing only by a rotation of the entire circle are identified. Fix a direction of reading around the circle; reflection, which reverses the order, is not identified. No numbered seat or external position distinguishes any seat.

Fix a particular person $A$ as a reference. Each circular order has exactly one representation beginning with $A$ in the chosen direction; the remaining $n-1$ people are then arranged in an order. Thus for $n\ge1$, the number of circular permutations is

This argument also holds for $n=1$, yielding $0!=1$. If seats are labeled or positions distinguished, moving $A$ to two different seats produces different outcomes; there are then $n!$ arrangements even if the seats form a circle. “Identical seats” leads to the circular formula only if rotations of the entire arrangement are also identified.

3. Worked examples

Example 1. Three people in a row and around a circle

Three friends $A,B,C$ can occupy three positions in a row in $3!=6$ ways. Read from left to right, these are

ABC, ACB, BAC, BCA, CAB, CBA.

If three seats are equally spaced around a circle and whole-circle rotations are identified, a clockwise rotation through $120^\circ$ does not create a new circular order. The six lists form two rotation classes: $ABC,BCA,CAB$ and $ACB,CBA,BAC$.

Fix $A$ as the starting point and read clockwise. The two representatives are $ABC$ and $ACB$, so there are $2!=(3-1)!=2$ orders. Reverse reading orders are not automatically identified by reflection.

Example 2. Four people with only circular order distinguished

Arrange $A,B,C,D$ around four unnumbered seats, identifying rotations. Fix $A$ and arrange $B,C,D$ clockwise. There are $3!=6$ ways. Fixing $A$ merely selects one representative of each circular order; it does not add a labeled seat to the problem.

Example 3. Four numbered seats

Use the same four people, but number the seats from $1$ to $4$. Placing $A$ in seat $1$ differs from placing $A$ in seat $2$. Filling seats $1,2,3,4$ successively allows $4,3,2,1$ choices, respectively, so the total is

Alternatively, take the $6$ circular orders in Example 2 and choose one of $4$ seats for $A$: $6\cdot4=24$. This determines the other three seats uniquely from the chosen order.

4. Practice exercises

1. In how many ways can $5$ distinct students occupy $5$ seats in a row?
2. Arrange $5$ people around a round table, identifying rotations but distinguishing the two reading directions. How many arrangements are there?
3. Arrange $5$ distinct people in a row with $A,B$ adjacent. How many arrangements are there?

5. Common mistakes

• Using $(n-1)!$ merely because the table is round, without checking whether seats are labeled.
• Dividing again by $2$ even though the problem distinguishes reverse orders.
• Using $n!$ when selecting only some of the $n$ elements.
• Forgetting that elements must be distinct or allowing a person to occur more than once.

6. Related lessons

• The addition and multiplication principles (English lesson)
• Permutations and combinations (English lesson)
• The binomial theorem (English lesson)

Original exercise resource on Scribd.