A circular Permutation is the total number of ways in which n distinct objects can be arranged around a fixed circle. A permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word “permutation” also refers to the act or process of changing the linear order of an ordered set.

What is Circular Permutation?

Circular permutation is the total number of ways in which n distinct objects can be arranged around a fixed circle. In the circular permutation, there is nothing like a start or an end.

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Cases of Circular Permutation

There are 2 cases of circular permutations

• If clockwise and anti-clockwise orders are different, then a total number of circular permutations is given by \(P_n=(n-1)!\)
• If clock-wise and anti-clock-wise orders are taken as not different, the total number of circular permutations is given by \(P_n={(n-1)!\over{2!}}\)

The properties of circular permutation are somewhat similar to the properties of permutation. There are 2 types of Circular Permutation:

• If Clockwise and Anticlockwise orders are different.
• If Clockwise and Anticlockwise orders are the same.

In circular permutation, one element is always fixed and all other elements are arranged relative to the fixed element. There is nothing like a start or an end. There is nothing like a beginning or end in circular permutation.

Properties of Circular Permutation

• Rotation Doesn’t Change the Arrangement
  In a circle, rotating all the objects together doesn’t create a new arrangement. That’s why circular permutation counts fewer arrangements than linear permutation.
   

• Fixing One Object
  To avoid counting duplicate rotations, we fix one object’s position and arrange the rest. That’s why the formula is (n – 1)! instead of n!.
   

• Clockwise vs Counter-clockwise

If clockwise and counter-clockwise arrangements are considered the same (like in necklaces or bracelets), then the formula becomes:
(n – 1)! / 2

If direction matters (like seating at a round table), use (n – 1)! only.

• Identical Items
  If some items are repeated, divide by the factorial of the repeated items:
  (n – 1)! / (p! × q! × ...)
  where p, q… are the counts of repeated items.
   

• Useful in Real-Life Arrangements
  Circular permutation is commonly used in situations like:

Seating around a round table

Designing circular flower beds

Arranging beads in circular jewelry

Formula and Derivation of Circular Permutation

Let’s study the formula and derivation of circulation permutation with proof.

Formula 1. If Clockwise and Anticlockwise orders are different

When the clockwise and anticlockwise orders are different then: The number of circular permutations is.

\(P_n={nP_r\over{r}}\)

where n = Total number of objects.

r = Number of selected objects.

Pn = Circular permutation.

If n = r (selecting all the objects) then,
\(\begin{matrix}
P_n={nP_n\over{n}}\\
= {n\times(n−1)!\over{n}}\\
P_n=(n−1)!
\end{matrix}\)

Here the arrangement (1-2-3 and 1-3-2) is different when we see the images from clockwise and anticlockwise.

Proof of Circular Permutation if clockwise and anticlockwise orders are different

Suppose n things \((x_1, x_2, x_3,…, x_n)\) are to be arranged around in a circular fashion. There are n! ways in which they can be arranged in a row. On the other hand, all the linear arrangements depicted by

\(\begin{matrix}
x_1 , x_2 , x_3,…, x_n\\
x_n , x_1 , x_2, …. , x_{n − 1}\\
x_{n − 1} , x_n , x_1 , x_2 …. x_{n − 2}\\
……
x_2 , x_3 , x_4 , ….., x_1\\
\end{matrix}\)

will lead to the same arrangement for a circular table. Hence each circular arrangement corresponds to n linear arrangements (i.e. in a row). Hence the total number of circular arrangements of n persons is n!/n = (n − 1)!

Formula 2. If Clockwise and Anticlockwise orders are same

The number of circular permutation is \(P_n={nP_r\over{r}}\)

r = Number of selected objects.

Pn = Circular permutation.

If n = r (selecting all the objects) then,

\(\begin{matrix}
P_n={nP_n\over{2n}}\\
= {n\times(n−1)!\over{2n}}\\
P_n={(n−1)!\over{2}}\\
\end{matrix}\)

Circulation Permutations with Repetition

Let us determine the number of distinguishable permutations of the letters ELEMENT.

Let us determine the number of distinguishable permutations of the letters ELEMENT.

Suppose we make all the letters different by labeling the letters as follows: \(E_1LE_2ME_3NT\)

Now, all the letters are different from each other. In this case, there are (7-1)! = 6! different circular permutations are possible.

Let us consider one such case of circular permutation, where the elements are arranged in the order below:

\(LE_1ME_2NE_3T\)

We can form new arrangements of permutations from this arrangement by only moving the E’s. Clearly, there are 3! or 6 such arrangements. We list them below.

\(\begin{matrix}
LE_1ME_2NE_3\\
LE_1ME_3NE_2\\
LE_2ME_1NE_3T\\
LE_2ME_3NE_1T\\
LE_3ME_2NE_1T\\
LE_3ME_1NE_2T\\
\end{matrix}\)

Because the E’s are not different, there is only one arrangement LEMENET and not six. This is true for every permutation.

Arrangement in Circular Permutation

From the properties of permutations, we already know that four persons A, B, C, and D can arrange themselves in 4!

In the circular permutation, there is nothing like a start or an end.

In circular permutation, one element is always fixed and all other elements are arranged relative to the fixed element. The number of ways in which the other three persons arrange themselves when one of them has a fixed position is (4 − 1)! = 3! = 6.

Applications of Circular Permutation

The application of circular permutations is as follows:

• Protein engineers traditionally rely on amino acid substitutions to alter the functional properties of biomacromolecules, yet have largely overlooked the potential benefits of reorganizing the polypeptide chain of a protein by circular permutation (CP).
• A circular permutation is a relationship between proteins whereby the proteins have changed the order of amino acids in their peptide sequence. The result is a protein structure with different connectivity, but an overall similar three-dimensional (3D) shape.
• Circular Permutation is useful for making seating arrangements.

Solved Examples of Circular Permutation

Example 1: In how many ways can 6 men be seated around a circular table?

Solution: 6 men can be seated around a circular table in (6-1)! = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways.

Example 2: Find the number of ways in which 10 different beads can be arranged to form a necklace.

Solution: 10 different beads can be arranged in a necklace (circular form) =12×(10−1)!=12×(9)! ways.

Example 3: In a playground, 3 sisters and 8 other girls are playing together. In a particular game, how many ways can all the girls be seated in a circular order so that the three sisters are not seated together?

Solution: There are 3 sisters and 8 other girls in a total of 11 girls. The number of ways to arrange these 11 girls in a circular manner = (11– 1)! = 10!. These three sisters can now rearrange themselves in 3! ways. By the multiplication theorem, the number of ways that 3 sisters always come together in the arrangement = 8! × 3! Hence, the required number of ways in which the arrangement can take place if none of the 3 sisters is seated together: 10! – (8! × 3!) = 3628800 – (40320 * 6) = 3628800 – 241920 = 3386880.

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