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How to solve Circular Permutation Questions Quickly
Circular Permutation of an arrangement in a closed loop can be calculated by the formula ,(n-1) ! where n is the number of items.


Definition & How to Solve Circular Permutation
- Combination is an arrangement of objects where order does not matter.
- There are also arrangements in closed loops, called circular arrangements
Type 1: Find the greatest or smallest number.
Question 1. Anuradha invited her 5 friends for dinner. In how many ways she can make them sit around a circular table?
Options
- 120
- 12
- 24
- 72
Solution: Number of arrangements possible = (n − 1)!
= (5 – 1)!
= 4!
= 24
Correct option: 3
Question 2. 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?
Options
- 2520
- 2400
- 5040
- 720
Solution: Number of arrangements possible = (n − 1)!
= (7 – 1)!
= 6!
= 720
Correct option: 4
Question 3. In how many ways can 4 men and 4 women be seated at a circular table so that no 2 women sits together?
Options
- 414
- 120
- 240
- 144
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! * 4!
= 3 * 2 * 1 * 4 * 3 * 2* 1
= 6 * 24
= 144
Correct option: 4
Type 2: When clockwise and anticlockwise
Question 1. How many different garlands can be made using 10 flowers of different colors?
Options
- 181440
- 362880
- 145690
- 5040
Solution: Number of arrangements possible = \frac{1}{2} × (n-1) !
= \frac{1}{2} × (10-1) !
= \frac{1}{2} × 9 !
= \frac{1}{2} × 362880
= 181440
Correct option: 1
Question 2. How many necklace of 10 beads each can be made from 20 beads of different colors?
Options
- \frac{10!}{19^2}
- \frac{{19!}^2}{10!}
- \frac{19!}{19^2}
- \frac{10!}{10^2}
Solution: In case of necklace the clockwise or anticlockwise arrangements are not different. Therefore, the required ways
= \frac{^{20}P_{10}}{10 ×2}
= \frac{20! }{10! × 10 ×2}
= \frac{{19!}^2}{10!}
Correct Option : 2
Question 3. In how many ways can 7 different colors beads be threaded in a string?
Options
- 3600
- 450
- 360
- 540
Solution: As necklace can be turned over, clockwise and anti-clockwise arrangements are the same. Therefore, Number of arrangements possible
= \frac{1}{2} × (n-1)!
= \frac{1}{2} × (7-1)!
= \frac{1}{2} × 6!
= 360
Correct option: 3
Read Also – Formula to solve Circular Permutations questions
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