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. Since one item can be fixed and the others positioned around it, there are (n-1)! circular permutations of the ‘n’ objects. 5 persons seated around a circle-shaped table, for instance, produces 4! or 24 variants. It is important to distinguish between circular and linear permutations since the counting method is impacted by the arrangement’s cyclical nature. This idea offers a unique viewpoint on how to organize parts in a circular layout and has applications in a variety of industries, including seating arrangements, clock setups, and molecular structures.
What is a Circular Permutation?
Circular Permutations are a specific kind of permutation that we frequently encounter in both mathematics and everyday life. Let’s first define permutation before moving deeper into the subject. A permutation is a mathematical phrase that generally refers to the variety of arrangements that may be made with a given set of numbers. Permutation deals with the rearranging of the values if the set’s numerical values are already in ordered pairs.
A circular permutation is a permutation in which the numbers are arranged and/or rearranged in a circular pattern, with one element remaining fixed and the others being positioned around it using the fixed element as the scale. In other words, the other values in the set are arranged in relation to the fixed number.
Circular Permutation Definition
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.
For ‘n’ items, there are (n-1)! circular permutations possible. This idea is used in situations like seating individuals around a circle-shaped table or comprehending cyclic patterns in numerous academic areas.
Example of 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: Pn = (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.
Pn = (n – 1) !
⇒ Pn = (5 – 1) !
⇒ Pn = 4!
Step 3: Calculate 4! (4 factorial):
Pn = 4 x 3 x 2 x 1
⇒ Pn = 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.
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:
Pn = (n – 1)!
Where,
- Pn stands for a circular permutation,
- n stands for the number of objects
Derivation:
Pn = nPr/r
- n represents the total number of objects.
- r stands for the number of selected objects.
- Pn​ denotes the circular permutation.
If n = r (If you are selecting all the objects) then the formula simplifies to,
Pn=nPn/n
Pn=n×(n−1)!/n
Pn=(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:
Pn = (n – 1)! / 2!
Where,
- Pn stands for a circular permutation.
- n stands for the number of objects
Derivation:
The number of circular permutation is Pn = nPr/r
- n represents the total number of objects.
- r stands for the number of selected objects.
- Pn​ denotes the circular permutation.
If n = r (If you are selecting all the objects) then the formula simplifies to,
Pn = nPn/2n
⇒ Pn = n×(n−1)!/2n
⇒ Pn = (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.
Read More,
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.
Pn = (n – 1)! / 2!
⇒ Pn = (7 – 1)! / 2!
⇒ Pn = 6! / 2!
⇒ Pn = 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
Pn = (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
Pn = (n – 1)! / 2!
⇒ P6 = (6 – 1)! / 2!
⇒ P6 = 5! / 2!
⇒ P6 = 60
Circular Permutation: FAQs
1. What is a Circular Permutation?
A circular permutation is an arrangement of objects in a circular order. Unlike linear permutations, the first and last elements are considered adjacent in a circular arrangement.
2. How is Circular Permutation Different from Linear Permutation?
In linear permutations, the order matters, and the arrangement is in a straight line. In circular permutations, the objects are arranged in a circle, and the starting and ending points are considered adjacent.
3. How Many Circular Permutations are Possible with ‘n’ Objects?
For ‘n’ distinct objects, there are (n-1)! circular permutations. This is because the circular arrangement can be rotated to (n-1) different starting points.
4. What is the Formula for Circular Permutation?
The formula for circular permutations of ‘n’ distinct objects is (n-1)!.
5. Can Circular Permutations Include Repetition of Elements?
Yes, circular permutations can include repetition of elements. In such cases, the formula is modified based on the number of repetitions of each element.
6. How to Calculate Circular Permutations with Some Fixed Elements?
If there are ‘m’ fixed elements in a circular permutation of ‘n’ objects, then the number of circular permutations is (n-m-1)!.
7. Can We Use Circular Permutation in Real-life Situations?
Yes, circular permutations are applicable in various real-life scenarios, such as arranging people in a circular table, scheduling events in a circular timeline, etc.
8. What is the Difference Between Permutations and Combinations in a Circular Setting?
Permutations deal with the arrangement of objects in a specific order, while combinations only consider the selection of objects without regard to the order. In a circular setting, the order matters, so permutations are used.
9. Are there any Tricks or Shortcuts for Circular Permutation Problems?
Understanding the concept and practicing problems is the best way to tackle circular permutation questions. However, for specific cases, recognizing symmetry and using the (n-1)! formula can simplify calculations.
Share your thoughts in the comments
Please Login to comment...