Circular Permutations Calculator
Count round-table arrangements.
Result
How it is calculated
For n distinct people or objects placed around a circle, where turning the whole circle does not create a new arrangement, the count is (n − 1)! = (n − 1) × (n − 2) × … × 2 × 1. One item is fixed as a reference seat, and the other n − 1 are arranged in order relative to it. The tool multiplies the integers from 2 up to n − 1 and prints the exact whole number with thousands separators. The input must be a whole number from 1 to 20; 0, negatives, decimals such as 7.5, and anything above 20 give no result. For n = 1 and n = 2 the answer is 1, since a lone person has one seat and two people can only sit opposite each other. The count treats a mirror image as a different arrangement: sitting clockwise as A, B, C is not the same as A, C, B.
Worked example
Eight guests are seated at a round table. The linear count would be 8! = 40,320, but rotations of the same seating are identical, so we fix one guest and arrange the other seven. That gives (8 − 1)! = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 distinct seating plans. Dividing 40,320 by 8 gives the same number, since each circular seating appears in 8 rotations.
Common mistakes
- Using n! instead of (n − 1)!. For 8 guests that is 40,320 rather than 5,040, because it counts every rotation of the same seating as new.
- Applying it to identical or repeated items. The formula needs every person or object to be different; with repeats, you would divide by the factorials of the repeated counts.
- Counting a necklace or bracelet with it. A bracelet can be flipped over, so mirror images are the same and the count is about half of this result; this tool does not halve it.
- Leaving out a person who has a reserved seat. Count everyone, host included: with 8 people the answer is 7! = 5,040, and fixing the host's seat does not change that number.
- Entering a number above 20 or a fraction such as 7.5. The tool needs a whole number from 1 to 20 and shows no result outside that.
About this calculator
A circular permutation counts the distinct ways to arrange n distinct objects around a circle, where arrangements that differ only by rotation are considered the same. While n objects in a straight line can be ordered n! ways, fixing one object as a reference point to cancel out the rotational duplicates leaves only (n−1)! genuinely different circular arrangements — for six people around a round table, that is 5! = 120 seatings rather than 720.
This distinction matters wherever position is relative rather than absolute: event planners use it to count round-table seating arrangements, puzzle and combinatorics courses use it to illustrate why “linear” and “circular” counting problems diverge, and it shows up in scheduling problems like round-robin tournament rotations where only who sits next to whom matters, not an arbitrary starting seat.
This calculator applies the (n−1)! formula directly to your value of n, giving you the exact count of unique circular arrangements — and it’s worth noting this differs from “necklace” counting, which further divides by 2 when flipping (reflection) is also considered equivalent.
Was this helpful?
You can pick up to 3 reactions.
Popular calculators
Loan Calculator
Monthly payment, total interest and full amortization schedule.
Percentage Calculator
Percent change, percent of, and ratio calculations.
Discount Calculator
Calculate final price, savings, and discount amount.
BMI Calculator
Body mass index from height and weight (metric or imperial).
Age Calculator
Compute your exact age in years, months and days from a birth date.
Compound Interest Calculator
Calculate the final amount and interest earned with compounding.
All Calculators
No calculators found
Comments (0)