CIRCULAR PERMUTATION QUESTIONS AND LOGIC
- 2. 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.
- 3. 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!
- 4. 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.
- 5. 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.
- 6. 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
- 7. 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
- 8. • 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
- 9. 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
- 10. 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)!
- 11. • A necklace is made using 6 different colored beads. How many distinct necklaces can be formed by rearranging the beads? QUESTION 3
- 12. • 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
- 13. 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?
- 14. • 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
- 15. 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?
- 16. 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
- 17. 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
- 18. • 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
- 19. • Type 2: When clockwise and anticlockwise 11. How many different garlands can be made using 10 flowers of different colors? • 181440 • 362880 • 145690 • 5040
- 20. • 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
- 21. 13.In how many ways can 7 different colors beads be threaded in a string? • 3600 • 450 • 360 • 540
- 22. • 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. A) 12 B) 50 C) 100 D) 200 QUESTION 14
- 23. Determine the ways in which 4 married couples are seated on a round table if the spouses sit opposite to one another. A) 48 B) 36 C) 45 D) 60 QUESTION 15
- 24. • Calculate the number of ways in which 10 beads of a necklace can be arranged? a) 181440 b) 118400 c) 181404 d) 18400 QUESTION 16
- 25. • 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? a) 17 b) 24 c) 4 d) 5 QUESTION 17
- 26. • 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. A) 236541 B) 362880 C) 230145 D) None of these QUESTION 18
- 27. • 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
- 28. • 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
- 29. • 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
- 30. • 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
- 31. • Determine the ways in which 4 married couples are seated on a round table if the men and women must sit alternatively. a) 152 b) 144 c) 200 d) 235 QUESTION 23
- 32. THANK YOU
Editor's Notes
- #7: Number of arrangements possible = (5 − 1)! = 4! = 4 x 3 x 2 x 1 = 24 Correct option: 2
- #8: Correct option: 3 5 Married couples means we have to arrange 10 peoples First women can be placed anywhere in a circular Round Table and her husband in 1 way. Second women can be placed in 8 ways and her husband in 1 way. Third woman can be placed in 6 ways and her husband in 1 way. Fourth women can be placed in 4 ways and her husband in 1 way. Fifth women can be placed in 2 ways and her husband in 1 way. Total number of ways 1×8×1×6×1×4×1×2×1=384Ways
- #9: Correct option: 1 Since In formation of Necklace the Clockwise and Anti-clockwise Arrangements are Same So we divide by 2 Number of arrangements possible = 1221 × (n−1)! 1/2 × (8−1)! 1/2 × 7! 1/2 × 5040 = 2520
- #12: Solution : For a necklace, the arrangement is considered the same up to rotation. Total number of ways to arrange 6 different beads in a line is 6! (factorial), but we need to divide by 6 to account for the circular arrangement of the necklace. Hence, the total number of distinct necklaces = (6-1)! = 5! = 120
- #13: Solution: D Arrange 4 men sit at a round table= (n-1)! => 3! Now we have alternate 4 seats for Women => 4! No of ways = 3!*4! =144
- #14: Consider the captain and vice-captain as a single entity. we have 7 entities (the pair of captain and vice-captain and the other 6 players) to arrange around a circular table. Number of ways to arrange 7 entities in a circle is (7-1)! = 6! However, within the captain-vice-captain entity, there are 2 ways to arrange them. Hence, the total number of arrangements = 2 * 6! = 144 ways
- #15: Let’s assume each pair of book and pen together as a single entity. Now, we have 5 entities (5 pairs) to arrange around the circular shelf. Number of ways to arrange 5 entities in a circle is (5-1)! = 4! However, within each pair, there are 2 ways to arrange the book and pen. Hence, the total number of arrangements = 2^5×4!=32×24=768 ways
- #16: Let’s treat Alice and Bob as a single entity (AB) and David and Eve as a single entity (DE) Now, we have 4 entities (AB, Carol, DE) to arrange around the circular table. The number of ways to arrange 4 entities in a circle is (4-1)! = 3! *2 (AB entity)=12 ways However, within the AB entity, there are 2 ways to arrange Alice and Bob, and within the DE entity, there are 2 ways to arrange David and Eve and for c entity, there are 2 ways . 2*2*2 = 8 ways Hence, the total number of arrangements is = 12-8 = 4 ways
- #17: Number of arrangements possible = (n − 1)! = (5 – 1)! = 4! = 24 Correct option: 3
- #18: Number of arrangements possible = (n − 1)! = (7 – 1)! = 6! = 720 Correct option: 4
- #19: Solution: 4 men may be seated in 3! ways, leaving one seat empty. Then at remaining 4 seats, 4 women can sit in 4! ways. = 3! x 4! = 3 x 2 x 1 x 4 x 3 x 2 x 1 = 6 x 24 = 144 Correct option: 4
- #20: Number of arrangements possible = 1/2 × (n-1) ! = 1/2 × (10-1) ! = 1/2 × 9 ! = 1/2 × 362880 = 181440 Correct option: 1
- #21: Solution: In case of necklace the clockwise or anticlockwise arrangements are not different. Therefore, the required ways = 20𝑃10/10×2 = 20!/10!×10×2 = 19!/10! Now the beads can be arranged in the Circular Fashion in (20-1) = 19 ! ways Required number of ways = 19!/10!×19! = 19!^2/10! Correct Option : 2
- #22: As necklace can be turned over, clockwise and anti-clockwise arrangements are the same. Therefore, Number of arrangements possible = 1/2 × (n-1)! = 1/2 × (7-1)! = 1/2 × 6! = 360 Correct option: 3
- #23: ANS: A If we consider the grandparents as a single unit then externally we have 4 people to be arranged in a round table and so externally it is 3! And internally both grandparents are to be arranged so that is 2! So total is 3!x 2! = 12
- #24: ANS: A 4 Married couples That is 8 people Second women can be placed in 6 ways and her husband in 1 way. Third women can be placed in 4 ways and her husband in 1 way. Fourth women can be placed in 2 ways and her husband in 1 way. Total number of ways 6×4×2=48 Ways
- #25: Ans: a Let the first bead’s position be fixed. Now we have to arrange the remaining beads = 10 -1 = 9 beads. These remaining 14 beads can be arranged in ways 14P9 = 9! Also, the necklace has no dependency on the position of the beads, as they can be arranged in clockwise or anti clockwise direction. Therefore, the required number of ways = 1/2× (9!) =181440
- #26: Ans: b We have to calculate total circular permutation in this case when the arrangement of the people is not fixed on a round table. Total number of people who needs to sit on a table = 5. If we fix one person around whom these people (5-1) = 4 Number of ways these people can be arranged = 4! = 24
- #27: Ans: b Total bangles with Anita = 3 + 5 +2 = 10 bangles. The number of ways in which they can be arranged = (10-1)! = 9! = 362880 ways
- #28: Ans: b Arrangement in the inner circle: Number of people in the inner circle: 6 When we fix one person, 6-1 = 5! The remaining 5 children can be arranged in ways = 5P5 = 5! In the case of circular permutation, as the arrangement is not fixed, the total number of ways students can be arranged = 5!/2 Therefore, the arrangement of the inner circle = 5!/2= 60 Similarly, in the case of the outer circle, We fix one student from the group. Now, there are 11 people remaining who needs to be arranged in a circular formation. The number of ways these people can be arranged = 11P11 = 11! As it is circular and nondependent arrangement, the number of ways will be =11!/2 Arrangement = 11×10×9×8×7×6×5×4×3×2×1/2 = 19958400 Therefore, the final number of ways to arrange children in inner as well as outer circles = 19958400+ 60 19958460 ways
- #29: Ans: a When A and B needed to be seated together across all the arrangements, these two can be counted as one. so 5 people Therefore, we now need to arrange five people in a circle = 4! = 24 ways But A and B can further be seated in two ways which make their arrangement = 2! Hence, the final arrangement = 24 x 2 = 48 ways
- #30: Ans: d Here, we need to fix the position of three people- Harry and two other people. We will now treat than as a single unit. Units that need arrangement = 19 Also, there can be 2 ways in which the other people can sit on either side of the host in 2 ways. Therefore, the number of ways of their arrangement = 18! The total number of ways = 18! x 2
- #31: Ans: b Let us assume that 5 girls sit together making them a single unit. Now we need to arrange only boys and a unit of girls = 10 + 1 = 11 people. In the case of the circular arrangement, the needed number of ways 10! X 5!
- #32: Ans: b Here we fix the position of one lady and calculate the number of ways people can be seated. Seating of the ladies = 3! The number of ways men can be seated = 4! Therefore, number of ways man and women can be seated alternatively = = 3! x 4! = 144 ways