Circular Permutation

Circular Permutation is an arrangement notion in which the objects are arranged in a closed loop. The beginning and end points are ambiguous, in contrast to linear layouts. A circular permutation is a configuration of items or components where the starting and ending positions are flexible. It entails keeping track of all the possible arrangements of things around a closed loop.

Since one item can be fixed and the others positioned around it, there are (n-1)! circular permutations of the 'n' objects. For Example - 5 persons seated around a circle-shaped table, produces 4! or 24 variants.

This idea is used in situations like seating individuals around a circle-shaped table or comprehending cyclic patterns in numerous academic areas.

Circular Permutation Formula

Circular permutations occur in 2 cases.

Clockwise and Anti-Clockwise Order is Different
Clockwise and Anti-Clockwise Order is Identical

Formula for Clockwise and Anti-Clockwise [When Order is Different]

When the clockwise and anticlockwise arrangements of the numbers are identical after they have been organized, the formula yields the total number of potential circular permutations. So, the formula is:

P_n = (n - 1)!
Where,
P_n stands for a circular permutation,
n stands for the number of objects

Solved Example on Circular Permutation

Example: How many alternative configurations of 5 balls are feasible in a circle, assuming that the clockwise and anticlockwise layouts differ?

Solution: We will use the circular permutations formula to compute the number of alternative configurations since the balls are arranged in a circle with the requirement that the clockwise and anticlockwise arrangements are different.

Step 1: Understand the circular permutations formula: P_n = (n - 1) !
This formula computes the number of configurations in a circle when rotations are considered separate.

Step 2: Apply the formula to the problem: We need to place 5 balls in a circle.
P_n = (n - 1) !
⇒ P_n = (5 - 1) !
⇒ P_n = 4!

Step 3: Calculate 4! (4 factorial):
P_n = 4 x 3 x 2 x 1
⇒ P_n = 24

Step 4: Interpretation of the result:
As a result, 24 distinct configurations of 5 balls in a circle are feasible, assuming that the clockwise and anticlockwise layouts differ.

Derivation:

P_n= ⁿP_r/r
n represents the total number of objects.
r stands for the number of selected objects.
P_n denotes the circular permutation.
If n = r (If you are selecting all the objects) then the formula simplifies to,
P_n=nP_n/n
P_n=n×(n−1)!/n
P_n=(n−1)!

Formula for Clockwise and Anti-Clockwise [When Order is Identical]

The number of cyclic permutations is determined by the formula when there is no difference in the orders of the elements in the clockwise or anticlockwise directions, i.e., when both orders of the members of the set are identical. So, the formula is:

P_n = (n - 1)! / 2!
Where,
P_n stands for a circular permutation.
n stands for the number of objects

Derivation:

The number of circular permutation is P_n= ⁿP_r/r
n represents the total number of objects.
r stands for the number of selected objects.
P_n denotes the circular permutation.
If n = r (If you are selecting all the objects) then the formula simplifies to,
P_n= nP_n/2n
⇒ P_n= n×(n−1)!/2n
⇒ P_n= (n−1)!/2

Applications of Circular Permutation

The following is how circular permutations are applied:

Protein engineers often use amino acid replacements to change the functional characteristics of biomacromolecules, but they have largely ignored the potential advantages of rearranging a protein's polypeptide chain by circular permutation.
A circular permutation is a connection between proteins in which the amino acids in their peptide sequence have been rearranged. A protein structure with variable connections but a generally comparable three-dimensional shape is the end product.
Making seating configurations may be done using the circular permutation method.

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Solved Example of Circular Permutation

Example 1: How many different ways are there to arrange 8 guys around a circular table?
Solution:

Eight guys sit around a circular table = (8-1)! = 7!
⇒ Eight guys sit around a circular table = 7 x 6 x 5 x 4 x 3 x 2 x 1
⇒ Eight guys sit around a circular table = 5040 ways

Example 2: There should be seven valuable stones. Make sure that every stone is a diamond. Find out how many different configurations there are for these diamonds.
Solution:

Here, all the diamonds are the same in the predicament. This implies that it is impossible to tell whether the stones are arranged in a clockwise or anticlockwise fashion. Thus, in this case, we use the second calculation to determine how many different ways the stones may be stacked.
Here n = 7
Consequently, the formula provides the number of potential circular permutations.
P_n = (n - 1)! / 2!
⇒ P_n = (7 - 1)! / 2!
⇒ P_n = 6! / 2!
⇒ P_n = 360

Example 3: Calculate the circular permutation of 6 people seated around a round table while (i) If the anticlockwise and clockwise orders are different. (ii) If the anticlockwise and clockwise orders are the same.
Solution:

Case 1: If the anticlockwise and clockwise orders are different. Here n = 6. Use the Formula
P_n = (n - 1)!
⇒ P₆ = (6 - 1)!
⇒ P₆ = (5)!
⇒ P₆ = 5 x 4 x 3 x 2 x 1
⇒ P₆ = 120
Case 2: If the anticlockwise and clockwise orders are the same. Here n = 6. Use the Formula
P_n = (n - 1)! / 2!
⇒ P₆ = (6 - 1)! / 2!
⇒ P₆ = 5! / 2!
⇒ P₆ = 60

Practice Problems on Circular Permutation

Problem 1: How many different ways can 10 people be seated around a circular table?

Problem 2: Calculate the number of distinct configurations for 9 identical beads on a necklace.

Problem 3: Determine the number of ways to arrange 4 different colored balls in a circle, considering identical clockwise and anti-clockwise orders.

Problem 4: How many ways can 12 flowers be arranged in a circular garland, assuming the clockwise and anti-clockwise arrangements are identical?

Problem 5: Find the number of ways to seat 7 people around a table if two specific people must sit next to each other.