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In combinatorics, a circular permutation is an arrangement of objects around a circle rather than in a straight line. The key difference from a standard (linear) permutation is that in a circle, there is no fixed "first" or "last" position, which means some arrangements that look different in a line are actually identical around a circle. This article explains the theory behind circular permutations, shows how and why the formula differs from the linear case, and works through a range of examples.
Quick Recap: What are Permutations?
A permutation is an arrangement of objects where order matters. For example, the PIN code 4567 is a different permutation from 7654, even though both use the same digits.
The number of ways to arrange 
distinct objects in a line is:
If we are selecting 
objects from (without repetition), the number of permutations is:
When repetition is allowed and we select items from types, the count is simply 
.
Linear vs Circular Permutations
In a linear permutation, objects are arranged in a row. There is a clear first position and a clear last position, so every shift creates a new arrangement.
In a circular permutation, objects are arranged around a circle. Because there is no fixed starting point, rotating every object one place around the circle does not create a new arrangement — it is the same circle viewed from a different seat.
Why does this matter? Consider seating 4 people — A, B, C, D — around a circular table. The arrangements ABCD, BCDA, CDAB, and DABC all look identical when everyone is seated in a circle, because each is simply a rotation of the same configuration. In a line, these would count as 4 different arrangements. In a circle, they count as 1.
For objects in a line, every arrangement can be rotated into positions that all look the same in a circle. This means the number of distinct circular arrangements is times fewer than the number of linear arrangements.
The Circular Permutation Formula
Case 1: Clockwise and anticlockwise arrangements are DIFFERENT
This is the standard case and applies to most problems (for example, people seated around a table where each person has a distinct left neighbour and right neighbour).
Why 
? There are 
linear arrangements. Since each circular arrangement corresponds to rotations that are identical, we divide by :
Example: 7 students sit around a circular table. The number of distinct seating arrangements is:
Compare this with the linear case: 
. The circular count is exactly 7 times smaller, because each circular arrangement has 7 equivalent rotations.
Case 2: Clockwise and anticlockwise arrangements are the SAME
This case applies when the circle can be "flipped over" — for example, threading beads onto a necklace or bracelet. A necklace can be turned over, so a clockwise arrangement and its mirror image (anticlockwise) are considered the same.
We divide by an additional factor of 2 because each arrangement and its reflection are identical.
Example: 7 beads of different colours are threaded onto a bracelet. The number of distinct arrangements is:
Tip for deciding which formula to use: Ask yourself, "Can I flip the arrangement over and get the same thing?" If yes (necklaces, bracelets, keyrings), use 
. If no (people at a table, items on a turntable), use latex!
| Arrangement Type | Formula | When to Use |
|---|---|---|
| Linear (all n objects) | n! | Objects in a row or line |
| Circular (direction matters) | (n - 1)! | People at around a table |
| Circular (direction does not matter) |  | Necklaces, bracelets, keyrings |
The key insight is that a circle has no fixed starting point, so we always divide the linear count by . If the arrangement can also be flipped (as with a necklace), we divide by a further factor of 2.
Worked Examples
How many ways can 8 different coloured balls be arranged in a circle, if clockwise and anticlockwise arrangements are considered different?
Using the formula for circular permutations where direction matters:
There are 5,040 distinct circular arrangements.
How many ways can 10 people be seated around a circular dinner table?
Seating people around a table is a standard circular permutation where clockwise and anticlockwise arrangements are different (each person has a distinct left and right neighbour):
There are 362,880 distinct seating arrangements.
Five different charms are placed on a bracelet. How many distinct arrangements are possible?
A bracelet can be flipped over, so clockwise and anticlockwise arrangements are the same. We use the formula:
There are 12 distinct arrangements.
A family of 6 is seated around a circular table for dinner. The parents insist on sitting next to each other. How many seating arrangements are possible?
When two people must sit together, treat them as a single unit. This gives us 5 units to arrange around the circle (the parent-pair plus the 4 other family members):
However, within the parent-pair, the two parents can swap positions (mum on the left or dad on the left). This gives:
By the counting principle:
There are 48 possible seating arrangements.
A committee of 9 people sits around a circular table. Three specific members — Alice, Bob, and Charlie — must all sit together. How many arrangements are possible?
Treat Alice, Bob, and Charlie as a single block. This gives 
units to arrange around the circle:
Within the block, Alice, Bob, and Charlie can be arranged in:
Total arrangements:
There are 4,320 possible seating arrangements.
Eight diplomats from different countries sit around a circular table. Two particular diplomats refuse to sit next to each other. How many valid seating arrangements are there?
First, find the total number of circular arrangements without any restriction:
Next, find the number of arrangements where the two diplomats do sit next to each other. Treat them as a single block, giving 7 units around the circle:
The two diplomats can swap positions within their block:
Subtract the restricted arrangements from the total:
There are 3,600 valid seating arrangements.
12 different beads are to be arranged on a necklace. How many distinct necklaces can be formed?
A necklace can be rotated and flipped, so we use the formula where clockwise and anticlockwise arrangements are the same:
There are 19,958,400 distinct necklaces.
Four couples attend a dinner party and sit around a circular table. Each couple must sit together. How many seating arrangements are possible?
Treat each couple as a single block. This gives 4 blocks to arrange around the circle:
Within each couple, the two people can swap seats:
Since there are 4 couples:
latex^4 = 2^4 = 16[/latex]
Total arrangements:
There are 96 possible seating arrangements.
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Binomial theorem’s and permutation and combination’s
Hi there! Thanks for your comment! The topics of the binomial theorem, permutations, and combinations are all closely related — great concepts to explore together!
Is problem 5 correct?
The last part is 10! Divided by 4!5!.
Thanks for your comment! 😊 You’re right to double-check that step — it’s great to see such careful attention. In this case, the expression 10!/(4!5!) doesn’t apply to this particular problem, as it represents a combination rather than the arrangement described. The correct solution is shown in the explanation above, and we’ve double-checked that problem 5 is correct as written.
Thanks again for taking the time to point it out!
Amazing content
It really help a lot to me…keep it up..
I need some solutions for numbers
Exercises 1 to 12 have covered the different types of combinations.
Bravo for the exercises.
Thanksssss