Circular & necklace permutation calculator

How to use (3 steps)

Circular: rotations are the same. Necklace: rotations and flips are the same.

These two arrangements are the same after rotation.

Method: —

Value: —

Digits: —

Formula: —

Show the reasoning

Circular permutation: (n−1)! (fix one item because rotations are identical).
Necklace / bracelet permutation: (n−1)! / 2 for n ≥ 3 (and n = 1,2 are special cases → 1).

Example: circular seating for n=8 → 7! = 5040. Necklace with n=5 distinct beads → 4!/2 = 12.

Mixing up whether reflection should be treated as the same (bracelet) or different (circular table).
For necklace permutations, remember the n = 1,2 exceptions.
If positions are labeled (fixed seats), use n! instead of a circular formula.

What is the difference between circular and necklace permutations?

Circular permutations identify rotations. Necklace permutations identify both rotations and reflections (flips).

Why is the circular permutation formula (n−1)!?

Fix one item to break rotational symmetry, then arrange the remaining n−1 items.

When does the necklace result become (n−1)!/2?

For n ≥ 3, each arrangement and its reflection are the same necklace, so you divide by 2.

Why is the necklace result 1 when n = 2?

With 2 distinct beads on a loop, rotation and flipping do not create a new arrangement, so there is only one unique necklace.

What if seats have numbers (fixed positions)?

Then rotations are different, so the count is n! (use the factorial/permutation calculator).