Circular Permutation - JEE Important Topic

Circular permutation is a particular type of permutation that we often come across in Mathematics as well as in real life. Before we dive further into the topic, let us first understand what permutation is! A permutation is a technical term in Mathematics essentially referring to the number of ways a particular set of numbers can be arranged. Unlike combination, here the order or the arrangement of the numbers actually needs to exhibit a pattern or fulfil some criterion. 

If the numerical values in the set are already in ordered pairs, then permutation deals with the rearrangement of the values. A circular permutation is that permutation model where the numbers are being arranged and/or rearranged in a circular order (Imagine around a circle) with one element staying fixed and the others being positioned around it taking the fixed element as the scale, that is, the other values in the set are arranged relative to the fixed number. Circular permutations and combinations are extensively used in different fields. 

Clockwise and Anti-Clockwise Arrangement of Circular Permutation

A circle can basically be seen as a clock and the direction of arrangement of the numbers actually play a key role in deciding the number of possible circular permutations or arrangements that can be obtained from a specific set. 

• In cases where after the numbers have been arranged, the clockwise arrangement and the anti-clockwise arrangement of the numbers are the same, the total number of possible circular permutations is given by the formula, 

(n – 1) !. ( For n elements)

• In cases where there is no distinction between the orders of the elements either in the clockwise direction or in the anticlockwise direction that is both the orders of the elements of the set, are one and the same, the number of the possible number of cyclic permutations is given by the formula $\dfrac{(n-1)!}{2}$ ( For n elements)

Let us take the following examples to have a better understanding of the clockwise and anticlockwise arrangements of the elements of a given set in the case of circular permutation.

Example 1: Let there be 5 precious stones, one emerald, one ruby, one diamond, one sapphire and one jade in a necklace. The necklace is nothing but a simple circle and the five stones are to be arranged around it. Find out the possible number of arrangements in this case.

Solution:

Here as we can see, the necklace is circular in shape and therefore the possible number of arrangements required here can be obtained by implementing the concept of circular permutation. 

Here n = 5 

Since the stones are very different, there will be a distinct difference in the order of their arrangement when comparing the clockwise and anticlockwise directions. Therefore as the clockwise and anticlockwise orders can be distinguished we use the first circular permutation formula (n-1)!. 

Therefore the possible number of circular permutations that we can get here $=(5-1)!=4!=24$

Example 2: This time, let all the stones be identical diamonds. Find out the number of possible ways in which these diamonds can be arranged.

Solution:

Here, in the given situation all the diamonds are identical. This means that the clockwise and the anticlockwise order of the stones cannot be distinguished. Thus we apply the second formula here to find out in how many ways the stones can be arranged. 

Here n = 5.

Therefore the possible number of circular permutations is given by the formula

$\dfrac{(n-1)!}{2}=\dfrac{(5-1)!}{2}=\dfrac{4!}{2}=12$

From the two above-given circular permutation examples, we can see very clearly why the formulas get changed. When there are two separate orders being exhibited by the clockwise and anti-clockwise direction of a circular permutation, the number of possible arrangements is more, considering two distinct categories. On the other hand, if the order of arrangement cannot be distinguished the same order of arrangement remains for both directions, and the possible number of arrangements gets reduced. This is the main reason behind the existence of the two different cases. 

Circular Permutations with Restrictions

Circular permutation with restrictions refers to those specific cases where a certain number of things or values either have to be always included or always excluded. The circular permutation formulas related to this concept are given as follows:

• First instance: When n things are to be arranged taking r at a time such that a particular thing is always to be included in each of the arrangements is given by the formula $rP_{r-1}^{n-1}$

• Second instance: When n things are to be arranged taken r at a time such that a particular item gets excluded in each of the possible arrangements is given by the formula $P_{r-1}^{n-1}$

Examples Related to Circular Permutation

Example 1: 20 people are invited to a party. In how many ways can they be seated around a table such that 2 particular people sit on either side of the host?

Solution:

Total number of invitees = 20

Let us consider the host and two guests on either of his sides as one unit.

Therefore, the total number of units to be considered now = ( 20 – 2 +1) = 19

Number of possible ways in which the 19 units can be arranged around a circular table = ( 19 – 1) ! = 18!.

If the two guests interchange, then the total number of possible arrangements becomes twice.

Therefore, the total number of circular permutations that is possible here = 18! × 2.

Example 2:  In how many possible ways can 20 people sit around a round table with 9 chairs?

Solution:

As we can see here, 20 people sitting around a round table with 9 chairs is similar to the case of restricted permutation, where n elements or elements are taken at a time.

Here, 9 people must be included and at the same time, the order of the elements in the clockwise and then anticlockwise directions are the same.

Therefore, the total number of possible circular permutations $=\dfrac{P_{9}^{20}}{9}$. 

Conclusion

Circular permutation refers to the arrangement of elements of an ordered set around a circle or a circular structure. It could refer to the sitting arrangement around a round table or the number of diamonds in a necklace. If the order in the clockwise and then anticlockwise direction can be distinguished the formula is given by (n – 1)! and if the orders of the arrangement in the two directions cannot be distinguished, it is represented by the formula  $\dfrac{(n-1)!}{2}$.