Necklace Symmetry Permutations Without Repeatition

1. Necklace Symmetry Permutations refer to All Distinct Orientation Independent Circular Permutation of a given Set of Distinct Items such that Any One Item can occur Only Once in a given Orientation Independent Circular Permutation.
   
   Circular Permutations are called Orientation Independent if Every Clockwise Circular Permutation and their Corresponding Counter-Clockwise Circular Permutation are considered to be a Single Permutation.
   
   Therefore Total Count of Necklace Symmetry Permutations Without Repeatition of \(N\) Objects/Items Taking \(R\) Objects/Items at a time (where \(R\leq N\) and \(R\neq 2\) ) is Half the Count of Circular Symmetry Permutations Without Repeatition  \(={\Large\frac{P(N,R)}{2R}}={\Large\frac{N!}{2R(N-R)!}}\)   (when \(R\leq N\) and \(R\neq 2\) )   OR  \(={\Large\frac{(N-1)!}{2}}\)   (when \(R=N\) and \(R\neq 2\) )
   
   When \(R=2\), Count of Necklace Symmetry Permutations Without Repeatition is same as Count of Circular Symmetry Permutations Without Repeatition (since any Clockwise Circular Permutation is same as Counter-Clockwise Circular Permutation).