BodhiAI

Nov. 15, 2019
Circular Permutations

If we arrange the objects along a closed curve viz a circle, the permutations are known as circular permutations. In case of Linear permutations (which have two extremes: Left extremum and Right extremum) every arrangement has the beginning and the end but there is no beginning and end in case of Circular Permutation.

Thus, in a circular permutation, we consider one object as fixed and now the remaining objects are arranged in the same way as we arrange in linear arrangements.

Thus, the number of permutations of 'n' Distinct Objects is (n-1)!.

Important

In case of Circular Permutations, anti-clockwise and clockwise order of arrangements are considered ad distinct permutations.

Difference Between Clockwise and Anti-Clockwise Arrangements

The number of circular permutations of n distinct things in which clockwise and anti-clockwise arrangements give rise to different permutations is (n-1)!

i.e. (n-1)! Clockwise Arrangements + Anti-Clockwise Arrangements

Example: The number of permutations of 4 persons seated around the round table is (4-1)!=3!. Because with respect to the table, the clockwise and anti-clockwise arrangements are distinct.

If Anti-Clockwise and Clockwise order of arrangements are Not-Distinct i.e. Same

Example: Arrangements of Beads, Necklace, arrangements of flowers in Garland etc., then the number of circular permutations of n distinct items is

Problem Solving Trick

It Distinction can be made between the clockwise and anticlockwise arrangement of n different things around a circle, then the number of permutations are (n-1)!. If no distinction can be made between the clockwise and anti-clockwise arrangements of n different things around a circle, then the number of permutations is