Circular Symmetry Permutations Without Repeatition
Circular Symmetry 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.
Hence, the Count of Circular Symmetry Permutations Without Repeatition of \(N\) Number of Distinct Items is calculated as follows
\(1\) Circular Symmetry Permutation Without Repeatition of \(N\) Distinct Items = \(N\) Symmetry Independent Permutations Without Repeatition \(N\) Distinct Items
\(2\) Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items = \(2\times N\) Symmetry Independent Permutations Without Repeatition \(N\) Distinct Items
\(3\) Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items = \(3\times N\) Symmetry Independent Permutations Without Repeatition \(N\) Distinct Items
\(\vdots\)
\(N!\) Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items = \(N!\times N\) Symmetry Independent Permutations Without Repeatition of \(N\) Distinct Items
\(\therefore\) \(N!\) Symmetry Independent Permutations Without Repeatition of \(N\) Distinct Items = \({\Large\frac{N!}{N}} = (N-1)!\) Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items
Since the Count of Symmetry Independent Permutations Without Repeatition of \(N\) Distinct Items is \(N!\), therefore the Count of Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items is \((N-1)!\)
Similarly, the Count of Circular Symmetry Permutations Without Repeatition of \(N\) Number of Distinct Items taking \(R\) Distinct Items at a time (where \(R \leq N\)) is calculated as follows
\(1\) Circular Symmetry Permutation Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time = \(R\) Symmetry Independent Permutations Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time
\(2\) Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time = \(2\times R\) Symmetry Independent Permutations Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time
\(3\) Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time = \(3\times R\) Symmetry Independent Permutations Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time
\(\vdots\)
\({\Large \frac{N!}{(N-R)!}}\) Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time = \({\Large\frac{N!}{(N-R)!}} \times R\) Symmetry Independent Permutations Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time
\(\therefore\) \({\Large\frac{N!}{(N-R)!}}\) Symmetry Independent Permutations Without Repeatition of \(N\) Distinct Items taking R Items at a time = \({\Large\frac{N!}{R(N-R)!}}\) Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items taking R Items at a time
Since the Count of Symmetry Independent Permutations Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time is \(P(N,R)={\Large\frac{N!}{(N-R)!}}\), therefore the Count of Circular Symmetry Permutations Without Repeatition of \(N\) Distinct Items taking \(R\) Items at a time is \({\Large\frac{P(N,R)}{R}}={\Large\frac{N!}{R(N-R)!}}\).