Circular Permutation Concepts

1. Seating Around a Table

What is the question? These questions ask how many ways people or objects can be arranged around a circular table, where rotations are considered the same arrangement.

Formula:

$$(n-1)!$$

Example: In how many ways can 8 people sit around a round table?

Solution:

For 8 people:

$$(8-1)!=7!$$ $$7!=5040$$

Answer: 5040 ways

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2. Circular Arrangement with One Person Fixed

What is the question? A person may be fixed in a particular seat, and the remaining people are arranged around them.

Formula:

If one person is fixed:

$$(n-1)!$$

Example: A, B, C, D and E are sitting around a circular table. If A must sit in a fixed position, in how many ways can the remaining people sit?

Solution:

A is fixed.

So arrange the remaining 4 people:

$$4!=24$$

Answer: 24 ways

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3. Men and Women Sitting Alternately

What is the question? Men and women have to sit around a circular table in alternating positions.

Formula:

If there are $n$ men and $n$ women:

$$(n-1)!\times n!$$

First arrange the men around the table:

$$(n-1)!$$

Then arrange the women in the gaps:

$$n!$$

Example: In how many ways can 4 men and 4 women sit around a table so that men and women sit alternately?

Solution:

Arrange 4 men:

$$(4-1)!=3!=6$$

There are 4 gaps for the women:

M   M   M   M
  _   _   _   _

Arrange 4 women:

$$4!=24$$

Total:

$$6\times24=144$$

Answer: 144 ways

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4. Men and Women with No Two Women Together

What is the question? Arrange men and women around a table so that no two women sit next to each other.

Formula:

For $m$ men and $w$ women, where $w\le m$:

$$(m-1)!\times {m\choose w}\times w!$$

First arrange the men:

$$(m-1)!$$

Then choose gaps for the women:

$${m\choose w}$$

Then arrange the women:

$$w!$$

Example: In how many ways can 5 men and 3 women sit around a table such that no two women sit together?

Solution:

Arrange 5 men:

$$(5-1)!=4!=24$$

There are 5 gaps:

M   M   M   M   M
 _   _   _   _   _

Choose 3 gaps:

$${5\choose3}=10$$

Arrange 3 women:

$$3!=6$$

Total:

$$24\times10\times6=1440$$

Answer: 1440 ways

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5. Two Particular People Must Sit Together

What is the question? Two specific people must always sit next to each other.

Formula:

Treat the two people as one block.

For $n$ people:

$$(n-2)!\times2$$

The $2$ comes from the two possible orders inside the block:

AB
BA

Example: In how many ways can 6 people sit around a table if A and B must sit together?

Solution:

Treat A and B as one block.

So we have:

[AB], C, D, E, F

That is 5 units around a circle:

$$(5-1)!=4!$$

A and B can switch positions:

$$2!$$

Therefore:

$$4!\times2=24\times2=48$$

Answer: 48 ways

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6. Two Particular People Must Not Sit Together

What is the question? Two specific people must not be next to each other.

Formula:

\text{Required} =============== ## \text{Total arrangements} \text{Arrangements together}

Example: In how many ways can 6 people sit around a table if A and B cannot sit together?

Solution:

Total arrangements:

$$(6-1)!=5!=120$$

Arrangements where A and B sit together:

$$(6-2)!\times2$$ $$=4!\times2=48$$

Therefore:

$$120-48=72$$

Answer: 72 ways

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7. A Particular Person Must Sit Between Two Particular People

What is the question? A person must have two specified people sitting immediately on their two sides.

Formula:

Fix the required central person, then arrange the remaining people while accounting for the two possible orders of the neighbours.

Example: 6 people A, B, C, D, E and F sit around a table. A must sit between B and C. How many arrangements are possible?

Solution:

Fix A.

B and C must occupy the two seats beside A.

They can be arranged as:

B A C

or

C A B

So:

$$2$$

The remaining 3 people can be arranged in:

$$3!=6$$

Therefore:

$$2\times3!=12$$

Answer: 12 ways

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8. Couples Must Sit Together

What is the question? Several married couples must sit around a circular table with each husband-wife pair sitting together.

Formula:

Treat each couple as one block:

$$(n-1)!\times2^n$$

where $n$ = number of couples.

Example: In how many ways can 4 married couples sit around a table if each couple must sit together?

Solution:

Treat each couple as one unit.

So there are 4 units:

$$(4-1)!=3!$$

Within each couple, husband and wife can switch:

$$2^4$$

Therefore:

$$3!\times2^4$$ $$=6\times16=96$$

Answer: 96 ways

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9. Couples Must Not Sit Together

What is the question? Several married couples sit around a table, but no husband can sit next to his wife.

Idea:

This is usually solved using total arrangements minus arrangements where couples are together, and for multiple couples it becomes an inclusion-exclusion problem.

Example: For 3 couples, how many circular arrangements have no husband sitting next to his wife?

Solution:

Total arrangements:

$$(6-1)!=5!=120$$

Let:

A = Couple 1 sits together
B = Couple 2 sits together
C = Couple 3 sits together

Using inclusion-exclusion:

Total
- arrangements with 1 specified couple together
+ arrangements with 2 specified couples together
- arrangements with all 3 couples together

One specified couple together:

$$(6-2)!\times2=4!\times2=48$$

For 3 choices of couples:

$$3\times48=144$$

Two specified couples together:

(6-3)!\times2^2 =============== # 3!\times4 24

There are:

$${3\choose2}=3$$

choices:

$$3\times24=72$$

All three couples together:

$$(6-3-? )$$

More directly, treat the 3 couples as 3 circular blocks:

(3-1)!\times2^3 =============== # 2!\times8 16

Therefore:

$$120-144+72-16=32$$

Answer: 32 ways

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10. Necklace / Garland Arrangement

What is the question? Arrange different beads around a necklace or garland where clockwise and anticlockwise arrangements are considered the same.

Formula:

$$\frac{(n-1)!}{2}$$

This is different from an ordinary round table because a necklace can be flipped.

Example: In how many ways can 6 different beads be arranged into a necklace?

Solution:

Circular arrangements:

$$(6-1)!=5!$$

Since clockwise and anticlockwise arrangements are identical:

\frac{5!}{2} ============ # \frac{120}{2} 60

Answer: 60 ways

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11. Clockwise and Anticlockwise Considered Different

What is the question? Sometimes the arrangement has a fixed direction, such as people sitting at a table where clockwise and anticlockwise orders are considered different.

Formula:

$$(n-1)!$$

Do not divide by 2.

Example: 7 people sit around a circular table. Clockwise and anticlockwise arrangements are considered different. Find the number of arrangements.

Solution:

$$(7-1)!=6!$$ $$=720$$

Answer: 720 ways

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12. Clockwise and Anticlockwise Considered the Same

What is the question? The arrangement can be flipped, so clockwise and anticlockwise orders represent the same arrangement.

Formula:

$$\frac{(n-1)!}{2}$$

Example: 7 different objects are arranged around a circle, and clockwise and anticlockwise arrangements are considered identical. Find the number of arrangements.

Solution:

\frac{(7-1)!}{2} ================ # \frac{6!}{2} 360

Answer: 360 ways

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13. People Sitting in a Row vs Around a Circle

What is the question? The question may look similar, but you must first identify whether the arrangement is linear or circular.

Formula:

For $n$ distinct people in a row:

$$n!$$

For $n$ distinct people around a circle:

$$(n-1)!$$

Example: 6 people can sit in a row in how many ways?

Solution:

Since it is a row:

$$6!=720$$

If the same 6 people sit around a circular table:

$$(6-1)!=120$$

Answer:

Row     → 720
Circle  → 120

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14. Seating with a Fixed Person and a Condition

What is the question? One person is fixed in a particular position, while another seating condition must also be satisfied.

Formula:

Fix the specified person first, then arrange the remaining people according to the condition.

Example: 6 people sit around a table. A is fixed at one position and B must sit next to A. How many arrangements are possible?

Solution:

A is fixed.

B can sit on either side:

B A

or

A B

So:

$$2$$

The remaining 4 people can be arranged:

$$4!=24$$

Therefore:

$$2\times24=48$$

Answer: 48 ways