Calculator
Use the group size field only when you choose the group-based scenario. Other scenarios ignore m automatically.
Growth Chart
Example Data Table
| n | Standard circular count | Mirror-same count | Adjacent-pair count |
|---|---|---|---|
| 3 | 2 | 1 | 2 |
| 4 | 6 | 3 | 4 |
| 5 | 24 | 12 | 12 |
| 6 | 120 | 60 | 48 |
| 7 | 720 | 360 | 240 |
| 8 | 5040 | 2520 | 1440 |
Formula Used
1) Standard circular permutations
For n distinct objects around a circle, only rotation is ignored.
2) Mirror images counted as identical
When reflection and rotation are both ignored, divide the standard count by 2 for n > 2.
3) Two specified objects adjacent
Bundle the pair into one block, then account for the two internal orders.
4) Two specified objects not adjacent
Subtract adjacent arrangements from the standard circular total.
5) A chosen group of m objects stays together
Treat the group as one block, arrange the blocks circularly, then arrange inside the group.
6) Two specified objects opposite each other
Valid for even n only. Fix one chosen object, place the other opposite it, then arrange the rest.
How to Use This Calculator
- Enter the total number of distinct objects in the circle.
- Select the arrangement rule you want to evaluate.
- Enter a group size only for the group-together case.
- Press Calculate to show the result above the form.
- Review the comparison table, chart, and exact count.
- Use the download buttons to export CSV or PDF.
FAQs
What is a circular permutation?
A circular permutation counts arrangements around a circle where rotations are considered identical. One position can be fixed as a reference, leaving the remaining objects to be arranged.
Why is the basic formula (n - 1)! instead of n!?
In a circle, rotating the whole arrangement does not create a new pattern. Fixing one object removes those repeated rotations, so only the remaining n - 1 objects are permuted.
When should mirror images be treated as the same?
Use that option for necklace-like or flip-equivalent settings. In those cases, clockwise and counterclockwise versions represent the same arrangement, so the count is halved for n greater than 2.
How does the adjacent-pair formula work?
Two chosen objects are grouped into one block. That reduces the circular problem by one effective item, then the pair can swap internally in two ways.
What does “not adjacent” mean here?
It means the two specified objects never sit next to each other. The calculator finds this by subtracting all adjacent cases from the full circular count.
When is the opposite-seat scenario valid?
That case only works when the total number of positions is even. With an odd number of objects, there is no single seat exactly opposite another seat.
Can this page handle large values of n?
Yes. It uses arbitrary-precision arithmetic for exact factorial-based results, so large counts remain accurate even when normal integer types would overflow.
Why does the chart use log10 values?
Factorials grow extremely fast. A logarithmic scale keeps the graph readable and still shows how each scenario expands as n increases.