Circular Permutation, Illustrations And Examples, Important Topics For JEE 2024

Circular Permutation : Permutation Gives possible arrangements of objects these could be categorized in two parts such as linear permutation and circular permutation core concept behind both of these arrangements is always visualize the concept with respect to the object means if we have to arrange a, b, c linearly than possible arrangements would be abc, acb, bac, bca, cab, cba, if someone would ask a about its position than it would describe its position as

• abc- at rightmost position and right to b
• acb- at rightmost position and right to c
• bac- at right to c and left to b
• bca- at leftmost position and left to c
• cab- at right to b and left to c
• cba- at leftmost position and left to b

Now if we have to arrange in circular manner than possible arrangement would be

Possible arrangements would be two only as abc, acb this is because

• abc- a is at right to b and left to c
• acb- a is at right to c and left to b
• bac- a is at right to c and left to b
• bca- a is at right to b and left to c
• cab- a is at right to b and left to c
• cba- a is at right to c and left to b

from above we could observe (acb, bac, cba) and (abc, bca, cab) represents two different arrangements

This arrangement could be written as 3-1 ! also means fix first position and arrange remaining values with respect to it so for n objects this could be written as n-1 !

Circular Permutation  Introduction

As we have discussed formula for circular arrangement is n-1 ! This formula is valid only when point of reference is not available for example if seats are marked with number and we have to arrange n peoples on it around a round circular table than number of ways would be n! this is because now everyone has a point of reference.

So, we can understand the scenarios when circular permutation would convert to linear permutation.

1. When seats are marked or only one seat is marked.

2. When size of seats varies or only one seat size varies.

Permutation between people around circular table would be considered as linear permutation, for example if we have to arrange 6 peoples around a table such that 2 particular persons are never together than permutation of these 2 peoples in gaps created would be as linear permutation, let’s explore some examples.

Examples based on circular permutation

circular permutation Example 1 :

Solution: Since persons has to sit around a round table than number of ways possible would be 6-1 !=120

circular permutation Example 2 :

Ram invited his 7 friends on his birthday in how many ways he can arrange them in circular table while Anand and Satyam do not want to sit together?

Solution: Let’s arrange remaining peoples around the table as in (5-1)! Ways now since Anand and Satyam do not want to sit together than they must be in the gaps created, gaps available are 5 and arrangement would be 25c×2! , Total ways are 4! 25c×2!=480

Circular permutation Rapid Questions

1. Sweta invited her 5 friends on her birthday in how many ways she can arrange them in circular table while Aparna and Priyanka do not want to sit together?

2. Find possible number of ways in which 8 persons can sit around a round table?

Complex Illustrations based on circular permutation

1. Find number of ways in which 6 men and 6 women can sit around a round table so that no men are adjacent to each other?

Solution: Here we have 6 men and 6 women available. First arrange all women in (6-1)! Ways now men can be arranged in gaps created in 6! total number of ways possible would be 5!×6!=86400

Circular permutation Rapid Questions

1. Find number of ways in which 8 men and 8 women can sit around a round table so that no women are adjacent to each other?

2. Find number of ways in which 5 men and 4 women can sit around a round table so that all women are not together?