Section 9.3 Circular Permutations
Subsection 9.3.1
Consider a necklace with \(n\) distinct beads. How many ways can the beads be arranged? Here, we consider two necklaces to be the same if one can be transformed to the other by a rotation. There are \(n!\) ways to order \(n\) distinct objects. However, for each of these permutations, there are \(n\) possible equivalent rotations. Thus, the number of necklace arrangements is,
\begin{equation*}
\frac{n!}{n} = (n-1)!
\end{equation*}
Suppose further that two necklaces are considered the same if they can be reflected to another (which they can, in real life). For each of these \((n-1)!\) arrangements, there are 2 equivalent reflections. Thus,
Theorem 9.3.1.
The number of distinct necklaces arrangements that can be formed with \(n\) distinct beads is,
\begin{equation*}
\frac{(n-1)!}{2}
\end{equation*}