Circular Permutations
Simple Explanation
When n distinct objects are arranged in a CIRCLE rather than a row, rotating the whole arrangement does not create a genuinely new arrangement — so the count is (n−1)!, not the usual n!.
Why Do We Need It?
Circular arrangements (like seating people around a table) are common enough to deserve their own formula, correcting for the rotational symmetry that a straight-line arrangement does not have.
Formula
Circular Permutations
(n−1)!
The number of distinct ways to arrange n distinct objects around a circle, where rotations of the same arrangement are considered identical.
- n
- — the number of distinct objects arranged around the circle
When to use it: Whenever objects are arranged in a circle (like people around a table) rather than in a row.
Worked Example
Count circular arrangements
In how many distinct ways can 6 people be seated around a circular table (where rotations of the same arrangement are considered identical)?
Why Does This Work?
Fixing one person's seat as a reference point removes the rotational duplication entirely — the remaining n−1 people can then be arranged in the remaining n−1 seats in any order, exactly (n−1)! ways.
Real-Life Example
Planning a seating chart for a round-table meeting
An event planner needs to know how many genuinely different seating arrangements are possible for guests around a circular table.
Since rotating everyone by one seat does not create a new arrangement (everyone's neighbors stay the same), the circular permutations formula gives the correct count.
Practice
In how many distinct ways can 5 people be seated around a circular table?
MediumCommon mistake
Using the regular n! formula instead of (n−1)! for circular arrangements — this forgets that rotations of the same arrangement are considered identical around a circle, unlike in a straight line.
Quick Review
- Circular permutations of n distinct objects = (n−1)!.
- Fixing one object's position removes the rotational duplication.
- Different from straight-line permutations (n!), which have no such symmetry to correct for.