Circular Permutations Without Repeatition refer to All Distinct Cyclic Arrangements of a given Set of Distinct Items
such that Any One Item can occur Only Once in a given Cyclic Arrangement.
Given \(N\) Number of Distinct Items , the Count of Circular Permutations Without Repeatition for \(R\) Items taken from \(N\) Items (where \(R \leq N\)) is calculated as follows
Number of Linear Permutations/Arrangements Possible for \(N\) Distinct Items taken \(R\) Items at a time= \(P(N,R)\) =\(\frac{N!}{(N-R)!}\)
Length of any Linear Permutation/Arrangement of \(R\) Distinct Items = \(R\)
\(\therefore\) Number of Cyclic Arrangements Possible for any one Linear Permutation/Arrangement of \(R\) Distinct Items = \(R\)
\(\therefore\) Number of Linear Permutations that Get Merged into Single Cyclic Permutation/Arrangement as they Represent the Same Cycle = \(R\)
Hence, Total Number of Distinct Cyclic Arrangements Possible for \(N\) Distinct Items taken \(R\) Items at a time =
Circular Permutations are called Orientation Independent if Every Clockwise Cricular Arrangement and their Corresponding Counter-Clockwise Circular Arrangement are considered to be a Single Permutation.
Since Every 2 Permutations are considered to be a Single Permutation , therefore
Total Orientation Independent Circular Permutations Without Repeatition \(=\frac{P(N,R)}{2R}=\frac{N!}{2R(N-R)!}\) (when \(R\leq N\))
OR Total Orientation Independent Circular Permutations Without Repeatition \(=\frac{N!}{2N(N-N)!}=\frac{(N-1)!}{2}\) (when \(R=N\))