Overview

Circular permutation problems involve arranging objects around a circle.

Circular arrangements are different from linear arrangements because rotations do not create new arrangements.

For example, if everyone at a round table shifts one seat clockwise, the seating is considered the same.

This symmetry is the main idea behind circular permutations.

Motivation

Suppose $4$ people sit around a round table.

In a row, they can be arranged in

ways.

But around a circle, rotations are considered the same.

For example,

around a table is the same circular arrangement as

Each circular arrangement has been counted $4$ times in the linear count.

So the number of circular arrangements is

In general, the number of ways to arrange $n$ distinct objects around a circle is

Fixing One Object

The easiest way to count circular arrangements is to fix one object in place.

Once one object is fixed, the rotational symmetry is removed.

Then arrange the remaining

objects around it.

This gives

Example 1

How many ways are there to seat $6$ people around a round table?

Fix one person.

Arrange the remaining $5$ people:

So there are $120$ circular arrangements.

Clockwise and Anticlockwise Arrangements

For ordinary round-table seating, clockwise and anticlockwise arrangements are usually considered different.

However, for necklaces or bracelets, flipping the object over may produce the same arrangement.

This depends on the problem statement.

If reflections are considered the same, then for $n>2$ distinct objects, the number of circular arrangements is

Only divide by $2$ when the problem clearly says that mirror images are the same, or when the object can physically be flipped.

People Sitting Together

If certain people must sit together, treat them as a block.

Example 2

How many ways are there to seat $6$ people around a round table if Alice and Ben must sit together?

Treat Alice and Ben as one block.

Now there are $5$ circular units:

• The Alice Ben block,
• The other $4$ people.

The number of circular arrangements of these $5$ units is

Inside the block, Alice and Ben can be arranged in

ways.

So the total number of arrangements is

People Not Sitting Together

If two people must not sit together, it is often easiest to use complement counting.

Example 3

How many ways are there to seat $6$ people around a round table if Alice and Ben must not sit together?

First count all circular arrangements:

Now subtract arrangements where Alice and Ben sit together.

From Example 2, that number is

So the number of arrangements where Alice and Ben are not together is

Alternating Around a Circle

Some problems ask for alternating arrangements, such as boys and girls around a circle.

The standard method is:

1. Arrange one group around the circle
2. Place the other group in the gaps

Example 4

How many ways are there to seat $4$ boys and $4$ girls around a round table so that boys and girls alternate?

First arrange the $4$ boys around the circle:

This creates $4$ gaps between the boys.

Now arrange the $4$ girls in those gaps:

Therefore, the total number of arrangements is

Circular Arrangements With Repeated Objects

If objects are repeated, circular arrangements become more delicate.

For basic contest problems, repeated circular arrangements are usually handled using casework or by first fixing a position and then accounting for repeated objects.

Example 5

How many ways are there to arrange the letters $A,A,B,C$ around a circle?

Fix $B$ in one position to remove rotational symmetry.

Now arrange

in the remaining $3$ positions.

Since the two $A$'s are identical, the number of arrangements is

So there are $3$ circular arrangements.

Necklace-Type Problems

If objects are arranged on a necklace, then rotations and reflections are often both considered the same.

For example, the clockwise arrangement

may be considered the same as

because the necklace can be flipped over.

For $n$ distinct objects on a necklace, the number of arrangements is

for $n>2$.

Always read the problem carefully to decide whether reflection symmetry should be counted.

Worked Examples

Example 6

How many ways are there to arrange $7$ people around a circular table?

Fix one person.

Arrange the remaining $6$:

Example 7

How many ways are there to arrange $7$ people around a circular table if three particular people must sit together?

Treat the three people as one block.

Then we have $5$ circular units, the block and the other $4$ people.

These can be arranged in

ways.

Inside the block, the three people can be arranged in

ways.

So the answer is

Example 8

How many ways are there to arrange $5$ distinct beads on a necklace?

Since this is a necklace, rotations and reflections are considered the same.

The number of arrangements is

Key Contest Insight

Circular permutations are about removing rotational symmetry.

For round table seating, fix one object and arrange the rest.

For necklaces or bracelets, check whether reflections are also considered the same.

Most circular restriction problems then become ordinary arrangement problems after the rotational symmetry is handled.

Strategy Tips

• For $n$ distinct objects around a circle, use $(n-1)!$
• Fix one object to remove rotational symmetry
• For people sitting together, use a block
• For people not sitting together, use complement counting
• For alternating arrangements, place one group first, then fill the gaps
• For necklaces, divide by $2$ if mirror images are considered the same
• Read the problem carefully to decide whether clockwise and anticlockwise arrangements are distinct.

Common Pitfalls

• Using $n!$ instead of $(n-1)!$ for circular arrangements
• Dividing by $2$ for round-table seating when reflections are not the same
• Forgetting internal arrangements inside a block
• Miscounting the number of gaps in alternating arrangements
• Treating necklaces and circular tables the same
• Forgetting repeated objects in circular arrangements

Practice Problems