Probability
Circular Permutations – Arrangements Around a Circle
Arranging n people in a straight line gives n! possible orders. But seat them around a circular table instead, and the count changes: rotating everyone around the table by one seat produces an arrangement that looks identical in terms of who sits next to whom. To account for this, circular arrangements are counted as (n − 1)! – exactly n times fewer than the straight-line count.
The reasoning is elegant: fix one person's seat as a reference point (removing the rotational duplicate-counting), and then arrange the remaining n − 1 people around them in the usual way.
The Circular Arrangement Formula
Circular permutations of n items = (n − 1)!, since rotating the whole circle produces n identical-looking arrangements for every truly distinct one.
(5 − 1)! = 4! = 4 × 3 × 2 × 1 = 24.
Real-Life Application
- Event seating: planning circular table arrangements at weddings or banquets.
- Group games: counting how many ways players can be arranged in a circle for a game.
- Necklace design: arranging beads of different colours around a circular necklace.
Key Takeaways
- Circular permutations of n items equal (n − 1)!.
- Rotations of the same circular arrangement are considered identical.
- Fixing one item's position removes the rotational over-counting.