CALCULATE WITH CONFIDENCE

Counting, Permutations & Combinations

Circular Permutation Calculator

Count arrangements of distinct objects around a fixed circle when rotations are equivalent.

Counting & Combinatorics

Arrange distinct objects around a circle

Rotations treated as equal
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example

Exact count

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Enter valid values to see the result.

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Understand the subject

What changes when arranging objects in a circle?

Around a circle, rotating everyone together does not change who sits next to whom. Fixing one object removes those duplicate rotations.

Use it for round tables and cyclic orders where rotations are equivalent.

The relationship

The counting rule

See the structure

Anchor one object to remove rotations

Worked example

Check the rule with a small case

Six people around a round table have (6−1)! = 120 relative arrangements.

Interpret with care

Choose the right counting model

Reflections are still different here. Necklace-style mirror equivalence needs another model.

Use the Permutation & Combination Generator when a small case needs to be inspected rather than only counted.

Quick guide

How to use this calculator

  1. Enter the number of distinct objects.
  2. Treat rotated copies as the same arrangement.
  3. Read the exact circular-permutation count.

Calculation method

Fix one object to remove rotational duplicates

Once one object is fixed as an anchor, the remaining n−1 distinct objects may be ordered freely.

Worked example

Seat six people around a table

Rotating everyone together does not create a new relative seating order.

(6−1)! = 120

Supported inputs

Precision and limits

Discrete domain

Inputs are whole counts. Negative, fractional, grouped, and scientific-notation inputs are rejected.

Exact integers

Results use arbitrary-precision integers and are never rounded. Inputs and output size are bounded to keep the page responsive.

Reflection

Clockwise and counterclockwise orders are distinct. Mirror-equivalent necklaces require a different model.