Calculation of permutations in a circular seating arrangement

In this post, we will study how the number of circular permutations from a different perspective. In the previous post, we studied circular permutations as the number of different arrangements that can be made of people sitting on a round table. In this post, we will learn how to calculate the number of different seating arrangements possible on a round table when a person does not have the same two neighbors.

There are five people A, B, C, D and E. They are seated on a table in two ways:

Both of the above arrangements are different, but in both the arrangements, each person has the same two neighbors. In the second arrangement, each person's neighbors on the left become their neighbors on the right and the neighbors at the right become the neighbors at the left. All the arrangements in which a person has the same two neighbors were included in the calculations shown in the previous post.

Here we will learn to calculate the number of circular permutations on the condition that each person should not have the same two neighbors.

Since for every circular permutation, there are two circular permutations in which a person will have the same two neighbors (as explained in the above diagram), therefore to obtain the total number of circular permutations in which no person has the same two neighbors, we will divide the total number of circular permutations by 2. Therefore,

Number of arrangement (permutations) in which no person has the same two neighbors

= (n - 1)! / 2

The above formula is also used to calculate the total number of different necklaces that can be formed from all different beads. This is because if we turn a necklace over, the arrangements of the beads becomes anticlockwise, and therefore there is no distinction between a clockwise and an anti clock wise direction in a necklace.