Master circular permutation with repetition through quick exact calculations. Review formulas, examples, and downloads easily. Useful for students, teachers, homework, and timed test practice.
| Case | Repeated Group Sizes | Total Items | Circular Count |
|---|---|---|---|
| A, A, B, B | 2, 2 | 4 | 2 |
| A, A, A, B, B | 3, 2 | 5 | 2 |
| A, A, B, C, C, C | 2, 1, 3 | 6 | 10 |
| A, A, B, B, C, C | 2, 2, 2 | 6 | 16 |
For repeated objects arranged on a circle, the exact count is based on Burnside’s Lemma.
Circular count = (1 / n) × Σ [φ(d) × (n / d)! / ((r1 / d)! (r2 / d)! ...)]
Here, n is the total number of items. The values r1, r2, ... are repeated group sizes. The divisor d runs through every divisor of the gcd of the repeated group sizes. The symbol φ(d) is Euler’s totient function.
When the gcd is 1, the formula reduces to the simpler circular multinomial result:
(n! / (r1! r2! ...)) / n
Circular permutation with repetition measures unique round arrangements when some items are identical. Rotation does not create a new arrangement. That rule changes the count. Repeated items also reduce the count because swapping identical objects changes nothing.
This calculator helps students, teachers, and exam learners solve ring arrangement problems quickly. It removes long manual factorial work. It also shows the reduced formula steps. That makes checking homework, quizzes, and competitive exam practice much easier.
For repeated circular arrangements, direct division is not always enough. Some patterns repeat under rotation. That symmetry must be counted correctly. Burnside’s Lemma handles this exactly and avoids errors in cases such as 2,2 or 2,2,2.
You may see this topic in necklace problems, seating puzzles, bead patterns, code wheel design, and round scheduling models. The same idea appears whenever position is circular and labels repeat. Understanding this model strengthens combinatorics, symmetry, and counting accuracy.
Enter repeated group sizes as comma separated integers. For example, 2,2,3 means three repeated groups with those sizes. The calculator adds them to get total objects. Optional labels simply make the explanation easier to read in the result panel and export file.
The tool returns total items, gcd, Burnside contributions, linear multinomial count, and circular count. It also shows an exact whole number and a scientific notation estimate. Use the example table below to compare your values and understand how repetition changes the answer.
A common mistake is using (n−1)! blindly. That only works in special cases. Another mistake is forgetting repeated symmetry patterns. Some learners also mix rotation with reflection. This calculator treats rotations as identical and counts mirror images separately unless a problem says otherwise.
Use this tool to test patterns, verify class notes, and improve speed. Try changing one group size at a time. You will see how symmetry affects the count. That simple practice builds intuition for permutations, multinomial reasoning, and circular arrangement questions.
It counts unique arrangements around a circle when some objects are identical. Rotations are treated as the same arrangement. Repeated items reduce the number of distinct outcomes.
Ordinary permutations treat positions in a line. Circular permutations treat rotated positions as identical. With repetition, identical items also remove duplicate arrangements, so the counting method changes again.
It fails when a repeated pattern stays unchanged after some rotation. In those cases, direct division misses extra symmetry. Burnside’s Lemma fixes that problem exactly.
The gcd helps identify which rotations can keep the pattern unchanged. Its divisors are used in the Burnside sum. If the gcd is 1, the problem becomes much simpler.
No. This calculator identifies rotations only. Reflections are counted separately. For necklace problems where flips also match, you would need a different symmetry model.
Yes. Labels are optional. They help you track which repeated group each count belongs to. The math uses the counts, while labels improve readability.
The table lists each divisor d, its totient value, the reduced totals, reduced counts, multinomial term, and contribution to the Burnside sum before the final division.
Use them when you want to save results, share solved examples, or keep a worksheet record. CSV is useful for data review. PDF is useful for printing.
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.