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Circular Permutation — Definition, Formula & Examples

A circular permutation is an arrangement of objects in a circle, where rotations of the same arrangement are considered identical. Because there is no fixed starting point, the number of distinct arrangements is fewer than in a straight line.

A circular permutation of nn distinct objects is an equivalence class of linear permutations under rotational shifts. Since each circular arrangement corresponds to exactly nn linear arrangements (one for each rotational position), the number of distinct circular permutations of nn objects is n!n=(n−1)!\dfrac{n!}{n} = (n-1)!.

Key Formula

Pcircular=(n−1)!P_{\text{circular}} = (n - 1)!
Where:
  • nn = The number of distinct objects arranged in a circle
  • (n−1)!(n-1)! = Factorial of (n − 1), the count of distinct circular arrangements

How It Works

In a linear permutation, the first position is fixed by the line itself—there is a clear left end and right end. In a circle, no position is inherently "first," so you can fix one object in place to eliminate duplicate rotations. Once that object is anchored, the remaining n−1n - 1 objects can be arranged in (n−1)!(n-1)! ways. If reflections (flipping the circle over) also count as identical—as with a bracelet or necklace—you divide again by 2, giving (n−1)!2\dfrac{(n-1)!}{2}.

Worked Example

Problem: In how many distinct ways can 6 people be seated around a circular table?
Identify n: There are 6 people, so n=6n = 6.
Apply the formula: Use the circular permutation formula (n−1)!(n-1)!.
(6−1)!=5!(6 - 1)! = 5!
Calculate: Compute 5!5!.
5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Answer: There are 120 distinct seating arrangements around the table.

Why It Matters

Circular permutations appear in scheduling round-table meetings, designing circular DNA sequences in biology, and arranging beads on a necklace. Many competition math problems (AMC, MATHCOUNTS) test whether you can distinguish circular from linear arrangements.

Common Mistakes

Mistake: Using n!n! instead of (n−1)!(n-1)! for circular arrangements.
Correction: Rotations of the same seating are identical in a circle. Fix one object's position to remove the nn rotational duplicates, which reduces n!n! to (n−1)!(n-1)!.