# Circular Permutations Questions and Answers

## Definition of Circular Permutations

On this page, you will get all the information related to Circular Permutation along with Sample Question.

Go through this page to get Sample Circular Permutation Questions and Answers for clear understanding of the Concept of Circular Permutation. ### Rules of Circular Permutations

Circular permutations can be a bit confusing as it completely different from the linear permutations or arrangement.

The below mentioned rules will help to give you insights regarding the rules as well as the formulas in order to avoid any mistakes.

• The numerous methods to organize different items along a stable (i.e., not able to chosen up out of the even and spun over) circle is Pn = (n-1)!.
• The number is (n-1)! as a substitute for the normal factorial n! as all cyclic arrangements of items are equal since the circle can be swapped.
• The total number of permutations decreases to $\frac{1}{2} (n-1)$ when there is no reliance identified.
• The same will be the situation when the location of the individual or thing does not rely on the arrangement of the permutation.
It is similar to the order of beads of a similar color in a necklace.

### Explanation for Circular Permutation using 5 Objects:

Circular permutation is a concept in combinatorics that deals with arranging objects in a circular or cyclic manner.

In a circular permutation, the order of objects matters, but rotations of the arrangement are considered identical.

To understand this better, let’s consider an example with five objects labeled A, B, C, D, and E arranged in a circle:

ABCDE

Now, if we were to rotate this arrangement, we get different “linear” permutations:

BCDEA, CDEAB, DEABC, EABCD

However, in circular permutation, these rotations are considered the same circular permutation because they represent the same arrangement when viewed in a circular manner. So, the circular permutations of the above arrangement are:

ABCDE (original arrangement)

BCDEA (rotation of original)

CDEAB (rotation of original)

DEABC (rotation of original)

EABCD (rotation of original)

In total, there are five circular permutations.

The formula to calculate the number of circular permutations for a set of n objects is (n-1)!.

This is because there is one linear arrangement for n objects, but we can rotate this arrangement in (n-1) ways without creating a new circular permutation.

It’s important to note that circular permutations are different from linear permutations, where the order of objects matters, and rotations are not considered identical.

In linear permutations, the number of arrangements for n objects is n!. However, in circular permutations, we divide the number of linear permutations by (n-1) to account for the identical rotations.

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Question 1 Time: 00:00:00
Find out the number of ways in which 5 members of a family can sit on a round table so that the grandparents always sit together. 12

12

50

50

100

100

200

200

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Question 2 Time: 00:00:00
Determine the ways in which 4 married couples are seated on a round table if the spouses sit opposite to one another. 48

48

36

36

45

45

60

60

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Question 3 Time: 00:00:00
Calculate the number of ways in which 10 beads of a necklace can be arranged? 181440

181440

118400

118400

181404

181404

18400

18400

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Question 4 Time: 00:00:00
Five people are sitting on a round table for meeting. These are P, Q, R, S, and T. In how many ways these people can be seated? 17

17

24

24

4

4

5

5

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Question 5 Time: 00:00:00
If Anita wants to arrange 3 Orange bangles, 5 Red bangles, and 2 Green bangles in a loop without any restrictions. Determine the number of ways it can be done. 236541

236541

362880

362880

230145

230145

None of the above

None of the above

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Question 6 Time: 00:00:00
A teacher needs to arrange the students of her classroom in two circles, one inside another. The inner-circle will have six members, and the outer circle will have 12 members. In how many ways these children can be arranged? 44545112

44545112

19958460

19958460

23569841

23569841

45789412

45789412

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Question 7 Time: 00:00:00
Determine the number of ways in which six people A, B, C, D, E, and F can be seated on a round table such that A and B always sit together. 48

48

120

120

300

300

320

320

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Question 8 Time: 00:00:00
Harry invited 20 people at a party. Determine the ways in which these people can be seated on a round table such that two specific people sit on either side of him. 20!

20!

16!

16!

18!

18!

18! x 2

18! x 2