Circular Permutations

CIRCULAR PERMUTATIONS

A circular permutation does not have a first or last position. The positions in the circle are relative to the other objects of the circle. Therefore, all of the following permutations are the same since B is opposite A, C is to the  right of A and D is to the left of A. Notice that all of the positions are given with respect to A. 

To find all the permutations of four people designated A, B, C and D above, person A, for example,  must walk to the designated area and begin the circle. As other people enter the circle, they have choices where to stand with respect to A. 

Let us assume that A starts the circle and the rest of the people enter the circle in alphabetical order. Hence, B has the above-mentioned three positions from which to choose. Once B has entered the circle, C has only 2 positions, relative to A, from which to choose. Once B and C have entered the circle, D has only one position remaining. 

Thus, the number of permutations of n objects in a circle is (n-1)!.

EXAMPLE 1: In how many ways can 8 people be seated around in a circular table?

(n-1)!

n = 8

(8-1)!

7! = 5040

EXAMPLE 2: In how many ways can 10 people be seated around a circular table if Donna and Kyle refuse to sit next to one another?

I. Calculate total.

(10-1)!

9! = 362880

II. Donna and Kyle together.

D + K = 2!

(9-1)! = 8!

(2!)(8!) = 2 (40320)

= 80640

III. Total - D+K together.

362880 - 80640 = 282240