circular permutations

Topics

• permutations and combinations

Circular permutation is the total number of ways in which n distinct objects can be arranged around a fixed circle. In the circular permutation, there is nothing like a start or an end.

There are 2 cases of circular permutations

• If clockwise and anti-clockwise orders are different, then a total number of circular permutations is given by $P=(n-1)!$
• If clock-wise and anti-clock-wise orders are taken as identical, the total number of circular permutations is given by $P=(n-1)!/2$

Proof

Suppose n things $(x_1,x_2,x_3,\dots ,x_n)$ are to be arranged around in a circular fashion. There are $n!$ ways in which they can be arranged in a row. On the other hand, all the linear arrangements depicted by

$$\begin{array}{rl}& x_1,x_2,\dots x_n\\ & x_n,x_1,\dots x_{n-1}\\ & x_{n-1},x_n,\dots x_{n-2}\\ & x_2,x_3,\dots x_1\end{array}$$

will lead to the same arrangement for a circular table. Hence each circular arrangement corresponds to $n$ linear arrangements (i.e. in a row). Hence the total number of circular arrangements of $n$ persons is $n!/n=(n-1)!$

For the case where the clock-wise and anti-clockwise orders are identical, by symmetry, we have the number as: $(n-1)!/2$

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