Circular Permutation Calculator
Calculate distinct arrangements of objects around a circle
Tip: In circular permutations, rotations of the same arrangement are considered identical.
Identical Objects
Understanding Circular Permutations
Basic Principle
For n distinct objects arranged around a circle, the number of unique arrangements is (n-1)!. We fix one object's position to account for rotational symmetry, then arrange the remaining (n-1) objects linearly.
Key Differences
- Linear permutations:
n! - Circular permutations:
(n-1)! - Rotated arrangements are identical
"Circular permutations reduce possible arrangements by a factor of n compared to linear permutations because rotations aren't counted as unique."
Practical Applications
Seating Arrangements
Arranging 5 people around a round table has (5-1)! = 24 unique arrangements (vs. 120 in a straight line).
Necklace Design
A necklace with 6 distinct beads has (6-1)!/2 = 60 unique designs (divided by 2 for mirror symmetry).
Note: For arrangements where mirror images are identical (like necklaces), divide by 2.
Formulas & Variations
Standard Formula
(n-1)!For n distinct objects arranged around a circle.
Identical Objects
(n-1)! / (k₁! × k₂! × ... × kₘ!)When there are identical objects (k₁ identical of type 1, etc.).
Frequently Asked Questions
Why (n-1)! instead of n!?
We fix one object's position to eliminate rotational duplicates. The remaining (n-1) objects are arranged linearly, giving (n-1)! unique arrangements.
How to handle identical objects?
Divide by the factorial of identical group sizes. Example: Arranging 3 red and 2 blue beads: (5-1)!/(3!×2!) = 4 unique arrangements.
When to divide by 2?
For arrangements where mirror images are identical (like necklaces or bracelets), divide the standard formula by 2.