CIRCULAR PERMUTATION QUESTIONS AND LOGIC

CIRCULAR PERMUTATION QUESTIONS

1. Circular Permutation Permutation arrangements are of many types, they range from linear to circular. Circular arrangements are the type of permutations where the people or things are organized in a circle. In other terms, this arrangement is said to be circular . Rules of Circular Permutations   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  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.

2. Tips and Tricks for Circular Permutation Circular Permutation are arrangements in the closed loops. If clockwise and anti-clock-wise orders are different, then total number of circular-permutations is given by (n-1)! If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by (𝑛−1)!/2!

3. Points To Remember: Choose a starting point: Even though the starting point doesn’t matter in circular arrangements, it can be helpful to pick a reference point to make the problem easier to solve. Think linearly: Sometimes, it’s easier to pretend the objects are in a straight line instead of a circle. Treat the ends as neighbors, and then use regular counting methods for linear arrangements. Treat groups together: If some objects must stay together or stay apart, treat them as one unit. This simplifies the problem and lets you consider arrangements of these groups and other individual objects separately.

4. Use (n-1)! for circles: For arranging n different objects in a circle, the number of ways is (n-1)!. This accounts for the circular arrangement and avoids over counting . Watch for repeated patterns: Rotations and reflections can create the same arrangements. Be careful and divide the total arrangements accordingly to avoid counting duplicates. Handle rules step-by-step: If there are specific rules or constraints, apply them thoughtfully. Break down the problem into smaller parts to deal with different scenarios if needed . Visualize the problem:  Draw pictures or imagine the circular arrangement to better understand the situation. Visualization can help you identify patterns and decide on the best approach. Practice with different situations:  Circular permutation questions can vary, so practice with various scenarios to get better at solving them.

5. Type 1: When clockwise and anticlockwise arrangements are different . Trick: Number of circular permutations (arrangements) of n distinct things when arrangements are different = (n−1)! In how many ways can 5 girls be seated in a circular order? A) 45 B) 24 C) 12 D) 120

6. QUESTION 1 Determine the number of ways in which 5 married couples are seated on a Circular Round table if the spouses sit opposite to one another. A) 120 B) 320 C) 384 D) 387

7. Type 2: When clockwise and anticlockwise arrangements are not different Tips & Trick: Number of circular permutations (arrangements) of n distinct things when arrangements are not different =  1/2​  × (n−1)! In how many ways can 8 beads can be arranged to form a necklace? 2520 5040 360 1200 QUESTION 2

8. Formula Number of circular-permutations of ‘n’ different things taken ‘r’ at a time:- Case 1:  If clock-wise and anti-clockwise orders are taken as different, then total number of circular-permutations =  𝑛𝑃𝑟/𝑟​​ Case 2:  If clock-wise and anti-clockwise orders are taken as not different, then total number of circular – permutation =  𝑛𝑃𝑟/2𝑟​​

9. Formulas & Definition for Circular Permutations: The arrangements we have considered so far are linear. There are also arrangements in closed loops, called circular arrangements. There are two cases of circular-permutations: If clockwise and anti-clock-wise orders are different, then total number of circular-permutations is given by (n-1)! If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by – ( 𝑛−1 )!/2!​ The number of ways to arrange distinct objects along a fixed (i.e., cannot be picked up out of the plane and turned over) circle is (n-1)!

10. A necklace is made using 6 different colored beads. How many distinct necklaces can be formed by rearranging the beads? QUESTION 3

11. In how many ways can 4 men and 4 women sit at a round table so that no two men are adjacent? (Two sitting arrangements are considered different only when the positions of the people are different relative to each other.) (A) 24 (B) 48 (C) 72 (D) 144 (E) 288 QUESTION 4

12. 5.A team of 8 basketball players is standing in a circle for a team photo. In how many ways can the players arrange themselves if the captain and vice-captain must stand next to each other?

13. In how many ways can 5 books and 4 pens be placed on a circular shelf such that no two pens are adjacent to each other? QUESTION 5

14. 7.Five friends – Alice, Bob, Carol, David, and Eve – are going to sit around a circular table. Alice and Bob want to sit together, but David and Eve cannot sit together. In how many ways can they be seated?

15. Type 1: Find the greatest or smallest number. Anuradha invited her 5 friends for dinner. In how many ways she can make them sit around a circular table? 120 12 24 72 QUESTION 8

16. 9. A gardener wants to plant some Neem trees around a circular pavement. He has 7 different size of Neem trees. In how many different ways can the Neem tree be planted? 2520 2400 5040 720

17. In how many ways can 4 men and 4 women be seated at a circular table so that no 2 women sits together? 414 120 240 144 QUESTION 10

18. Type 2: When clockwise and anticlockwise 11.  How many different garlands can be made using 10 flowers of different colors? 181440 362880 145690 5040

19. How many necklace of 10 beads each can be made from 20 beads of different colors? 10 !/19^2​ 19 !^2/10!​ 19 !/19^2 10 !/10^2 QUESTION 12

20. 13.In how many ways can 7 different colors beads be threaded in a string? 3600 450 360 540

21. 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 50 100 200 QUESTION 14

22. Determine the ways in which 4 married couples are seated on a round table if the spouses sit opposite to one another . 48 36 45 60 QUESTION 15

23. Calculate the number of ways in which 10 beads of a necklace can be arranged ? 181440 118400 181404 18400 QUESTION 16

24. 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 24 4 5 QUESTION 17

25. 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 362880 230145 None of these QUESTION 18

26. 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? a) 44545112 b) 19958460 c) 23569841 d) 45789412 QUESTION 19

27. 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. a) 48 b) 120 c) 300 d) 320 QUESTION 20

28. 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. a) 20 ! b) 16 ! c) 18 ! d)18 ! x 2 QUESTION 21

29. Determine the ways in which 5 girls and 10 boys can be seated on a table so that girls always sit together. a) 11 ! x 5! b) 10 ! x 5! c) 10 ! d) 5 ! QUESTION 22

30. Determine the ways in which 4 married couples are seated on a round table if the men and women must sit alternatively. 152 144 200 235 QUESTION 23

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