Solve necklace, table, and ring arrangements with confidence. Adjust duplicates, seats, and reflection rules quickly. Review totals, methods, exports, and examples in one place.
Choose a counting model, enter the required values, and calculate the number of unique circular arrangements.
| Scenario | Inputs | Formula / Method | Result |
|---|---|---|---|
| Distinct table seating | n = 6 | (6 - 1)! = 5! | 120 |
| Bracelet style counting | n = 6 | (6 - 1)! / 2 | 60 |
| Repeated symbols | n = 6, counts = 3,3 | Burnside’s lemma | 4 |
| Three items together, any order | n = 8, g = 3 | (8 - 3)! × 3! | 720 |
| Three items together, fixed order | n = 8, g = 3 | (8 - 3)! | 120 |
P = (n - 1)!
Fix one object as a reference point. Then arrange the remaining objects linearly around it.
P = (n - 1)! / 2 for n > 2
Use this when clockwise and counterclockwise layouts should count as the same arrangement.
P = (1/n) × Σ[ φ(d) × (n/d)! / Π(ri/d)! ]
This Burnside formula counts circular multisets exactly when some objects are identical. It avoids overcounting repeated rotational symmetries.
P = (n - g)! × g!
Treat the consecutive group as one block around the circle, then count all internal orderings of that block.
P = (n - g)!
The group is still treated as one block, but only one internal sequence is allowed.
In a circle, rotating all items together does not create a new arrangement. Linear ordering counts every position separately, but circular ordering removes equivalent rotations.
Use it when a clockwise arrangement and its mirror image should be treated as identical, such as some bracelet, ring, or reversible necklace problems.
Simple factorial division can fail when repeated counts create extra rotational symmetry. Burnside’s lemma handles those symmetries exactly and returns the correct circular count.
It means a chosen set of items must remain consecutive around the circle. The items can either be allowed to rearrange internally or kept in one fixed order.
Yes. It builds exact integer results using prime-factor methods and string arithmetic, so large whole-number outputs remain readable without floating-point rounding.
They must add up exactly to the total number of items. For example, if n is 8, counts such as 3,3,2 are valid.
Fixing one item removes duplicate rotations. It gives a stable reference point, which turns the remaining circular arrangement into a standard factorial count.
Yes. The CSV export creates a table-ready file, and the PDF button generates a formatted report you can save or share.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.