Linear permutations count n!
About this calculator
Linear permutations count n! orderings in a row. Around a circle, rotating everyone together does not change the seating — ABC, BCA, and CAB are the same round-table layout.
Fix one person to a reference seat. The remaining n−1 people arrange in (n−1)! ways. Six people around a table give 5! = 120 distinct seatings, not 720.
Enter n to compare circular (n−1)! with linear n!. For straight-line orderings use [[permutation-combination]]. For repeating symbols see [[permutations-with-replacement]].
Common uses
- Count round-table dinner seatings
- Solve necklace and bracelet counting problems
- Compare circular vs linear permutations in class
- Plan rotating shift schedules with fixed symmetry
Frequently asked questions
How many ways can 6 people sit at a round table?
(6−1)! = 5! = 120. Rotations that only spin the same seating do not count as new arrangements.
Why divide by n for circular permutations?
Each circular layout matches n different linear rotations. Fixing one seat removes that over-count, leaving (n−1)!.
What about flipping the circle over?
This calculator treats rotations as identical only. Reflections (mirror images) would divide by 2 more — a different problem.
Is (n−1)! the same as n!/n?
Yes. n!/n = (n−1)! when n ≥ 1. Both formulas count distinct circular orderings.