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Circular Permutation

When n distinct things are to be arranged in a straight line, we can do this in n! ways. However, if these n things are arranged in a circular manner, then the number of arrangements will not be n!.
 
Let us understand this:
The number of ways A, B and C can be arranged in a straight line = 3! = 6
 
The possible arrangements are — ABC, ACB, BAC, BCA, CAB, CBA
 
Now arrange these three people A, B and C in a circle
Description: 13-1.tif
 
What we observe here is that the arrangements ABC, BCA and CAB are the same. And similarly the arrangements ACB, CBA and BAC are the same.
 
So, there are only two permutations in this case of circular permutation.
 
To derive the formula for circular permutation, we first fix the position of one thing then the remaining (n − 1) things can be arranged in (n − 1)! ways.
 
Hence, the number of ways in which n distinct things can be arranged in a circular arrangement is (n – 1)!

It can be seen in the following way also:
If n things are arranged along a circle, then corresponding to each circular arrangement the number of linear arrangement = n
 
So, the number of linear arrangements of n different things = n × (number of circular arrangements of n different things)
 
Hence, the number of circular arrangements of n different things = (1/n) × number of linear arrangements of n different things = (1/n) × n! = (n – 1)!

Clockwise and Anti-clockwise Circular Arrangements

If we take the case of four distinct things A, B, C and D sitting around a circular table, then the two arrangements ABCD (in clockwise direction) and ADCB (the same order but in anti-clockwise direction) will be different and distinct. Hence, we can conclude that the clockwise and anti-clockwise arrangements are different. However, if we consider the circular arrangement of a necklace made of four precious stones A, B, C and D, the two arrangements discussed as above will be the same because we take one arrangement and turn the necklace around (front to back), then we get the other arrangement. Hence, the two arrangements will be considered as one arrangement because the order of stones is not changing with the change in the side of observation. So, in this case there is no difference between the clockwise and the anti-clockwise arrangements.
 
Summarizing the above discussion, the number of circular arrangements of n distinct things is (n−1)! if there is a difference between the clockwise and anti-clockwise arrangements and (n−1)!/2 if there is no difference between the clockwise and anti-clockwise arrangements.
 
Example
In how many ways 5 Indians and 4 Americans can be seated at a round table if
  1. There is no restriction
  2. All the four Americans sit together
  3. No two Americans sit together
  4. All the four Americans do not sit together
Solution
  1. Total number of persons = 9. These 9 persons can be arranged around a circular table in 8! ways.
  2. Assuming all the Americans to be one group, we have 6 things (5 Indians + 1 group) to be arranged around a circular table which can be arranged in 5! ways. However, these 4 Americans can be arranged in 4! ways among themselves.
     
    So, the total number of arrangements = 5! × 4!
  3. Since there is no restriction on Indians. The 5 Indians can be seated around a table in 4! ways. The Americans will now be seated between two Indians, i.e., 5 places. 4 Indians can be seated on these 5 places in 5P4 ways.
  4. The total number of arrangements when there is no restriction = 8! and the number of arrangements when all the four Americans sit together = 5! × 4!
     
    So, the total number of arrangements when all the four Americans do not sit together = 8! – 5! × 4!




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