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

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Question 9

Time: 00:00:00
Determine the ways in which 5 girls and 10 boys can be seated on a table so that girls always sit together.

11! x 5!

11! x 5!

10! x 5!

10! x 5!

10!

10!

5!

5!

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Question 10

Time: 00:00:00
Determine the ways in which 4 married couples are seated on a round table if the men and women must sit alternatively.

152

152

144

144

200

200

235

235

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Question 11

Time: 00:00:00
Find out the number of ways in which 3 people can be seated on a round table.

2

2

10

10

15

15

None of the above

None of the above

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Question 12

Time: 00:00:00
Calculate the number of garlands that can be made using 10 flowers ?

13125697

13125697

181440

181440

2362145

2362145

6987123

6987123

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Question 13

Time: 00:00:00
Determine the number of ways in which 20 people can be seated around a table if there are only 9 chairs.

$^{19}\textrm{C}_{9}\times 8!$

$^{19}\textrm{C}_{9}\times 8!$

$^{20}\textrm{C}_{9}\times 9!$

$^{20}\textrm{C}_{9}\times 9!$

$^{20}\textrm{C}_{9}\times 7!$

$^{20}\textrm{C}_{9}\times 7!$

$^{20}\textrm{C}_{9}\times 8!$

$^{20}\textrm{C}_{9}\times 8!$

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Question 14

Time: 00:00:00
5 Indian and 4 Americans need to be seated around a round table. In how many ways these people can be arranged so that all the Americans sit together?

1224

1224

2880

2880

3625

3625

2356

2356

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Question 15

Time: 00:00:00
When 4 Americans and 5 Indians are to be seated around a round table. Calculate the number of ways in which these people so that no two Americans sit together.

300

300

5! x 5!

5! x 5!

4! x 5!

4! x 5!

200

200

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Question 16

Time: 00:00:00
Find out the ways in which 5 boys and 4 girls can be seated on a round table such that all the 4 girls sit together.

236541

236541

17280

17280

12563

12563

None of the above

None of the above

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Question 17

Time: 00:00:00
Calculate the number of ways in which 8 ladies can be seated in a circle.

4569

4569

4056

4056

5040

5040

3625

3625

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Question 18

Time: 00:00:00
Determine the number of ways in which five people P, Q, R, S, and T can be seated around a round table in a way such that P and Q always sit together.

30

30

12

12

23

23

56

56

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Question 19

Time: 00:00:00
Calculate the number of ways in which P, Q, R, S, and T must be seated such that R and S must never sit together.

25

25

12

12

36

36

89

89

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Question 20

Time: 00:00:00
Determine the ways in which 3 Red and 3 Black bowls can be placed in a circle such that no 2 Black bowls are together.

14

14

20

20

56

56

12

12

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Question 21

Time: 00:00:00
Determine the number of ways in which 10 crystal beads can be arranged to make a bracelet.

181440

181440

123654

123654

181400

181400

181404

181404

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Question 22

Time: 00:00:00
Determine the number of ways in which the top 3 ranks can be given in a competition in which there are 9 participants?

504

504

300

300

550

550

450

450

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Question 23

Time: 00:00:00
Determine the number of ways in which 5 Blue and 4 Green bottles can be arranged in a circle without any restriction.

12569

12569

40320

40320

45621

45621

78912

78912

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Question 24

Time: 00:00:00
Alex invites 3 French and 2 German friends to her party. If she wants all of them to sit around a Round Table what are the number of ways this can be possible?

520

520

100

100

120

120

200

200

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Question 25

Time: 00:00:00
Find out the number of ways in which 7 English and 4 maths books can be arranged in a circular table such that all the 4 Maths books are kept together.

871023

871023

320145

320145

120960

120960

None of the above

None of the above

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Question 26

Time: 00:00:00
6 boys need to enter a boat which has 8 seats. If the boat has 4 seats on each side, determine the number of ways in which they can sit anywhere.

21203

21203

20160

20160

78450

78450

201369

201369

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Question 27

Time: 00:00:00
A boat has 8 seats in which 6 boys want to sit. If two boys P and Q want to sit on the port side and a boy R wants to sit on a starboard side, in how many ways they can be seated?

960

960

1000

1000

800

800

560

560

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Question 28

Time: 00:00:00
Calculate the number of ways in which five a, b, c, d, and e balls can be arranged in a circle.

30

30

24

24

50

50

80

80

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Question 29

Time: 00:00:00
There are 7 beads of different colors. Determine the ways in which these can be arranged in a string.

360

360

120

120

400

400

450

450

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Question 30

Time: 00:00:00
There are 3 Red, 5 Green, and 2 Black beads of different type. Determine the number of ways in which these can be arranged in a bracelet so that beads with similar color always stay together.

5469

5469

2880

2880

1230

1230

4561

4561

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