Mathematics and Statistics · Ch 15 — Permutations and Combinations
Circular Permutations
Circular Permutations
When objects are arranged around a circle (people around a round table, beads spaced on a ring), there is no fixed "first" position — rotating everyone one seat to the left gives the same circular arrangement. So each distinct circular arrangement corresponds to different linear arrangements (one for each starting point), and the linear count must be divided by .
Circular Permutations of Distinct Objects
Fix any one object's seat as a reference point, then arrange the remaining objects relative to it in ways.
Example: people can be seated around a round table in distinct ways.
When Clockwise and Anticlockwise Are the Same
If the arrangement can be flipped over so that a clockwise reading and its anticlockwise mirror image count as identical (typical for a garland of flowers or a necklace of beads), each arrangement is counted twice, so the number is
…
An arrangement of objects around a circle, where rotations of the same order are not counted as different: distinct objects give $(n-1) …
When the ring can be flipped so clockwise and anticlockwise readings are identical, the count is $\ …