Practice Questions
Concept Explanation
In a circular arrangement, rotations are considered identical. Fix one position to remove rotational symmetry.
Key Formulas
- n distinct objects around a circle = (n − 1)!
- Necklace/bracelet (flip allowed): (n − 1)! / 2
- Linear row of n distinct objects = n!
Example
6 people around a round table: (6 − 1)! = 5! = 120.
Common Trap
Do not use n! for circular seating; divide out rotations using (n − 1)!.
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Q1 EasyIn a campus seating drill, 6 distinct people sit around a round table. In how many different ways can they be seated if rotations are considered the same?ExplanationFor circular seating of n distinct people, arrangements = (n − 1)!. Here n = 6, so (6 − 1)! = 5! = 120. (Using 6! would count rotations as different.) Hence the answer is 120.
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Q2 EasyIn how many ways can 7 people sit around a circular table?ExplanationCircular arrangements of n distinct people = (n − 1)!. For n = 7: (7 − 1)! = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Hence the answer is 720.
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Q3 EasyFour distinct keys are placed on a circular keyring where only rotations are considered the same. How many distinct arrangements are possible?ExplanationCircular arrangements of 4 distinct keys = (4 − 1)! = 3! = 6. Hence the answer is 6.
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Q4 EasyThe number of ways to arrange n guests around a round table is:ExplanationIn a circular arrangement, rotations look identical, so one position is fixed and the remaining (n − 1) guests are arranged in (n − 1)! ways. Hence the answer is (n − 1)!.
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Q5 EasyThree friends sit around a small round table. In how many ways can they be seated?Explanation(3 − 1)! = 2! = 2. (The three linear permutations of the other two relative to a fixed person give only two distinct circular seatings.) Hence the answer is 2.
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Q6 EasyIn how many different ways can five persons sit around a circular table?ExplanationCircular seating of 5 persons = (5 − 1)! = 4! = 24. Hence the answer is 24.
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Q7 MediumEight executives sit around a conference table. Two particular executives insist on sitting next to each other. In how many ways can they be seated?ExplanationTreat the two who must sit together as one unit. Then 7 units sit in a circle in (7 − 1)! = 6! ways, and the pair can switch places in 2 ways. Total = 6! × 2 = 720 × 2 = 1440. Hence the answer is 1440.
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Q8 MediumFive people can sit in a row in 5! ways and around a circular table in (5 − 1)! ways. The ratio (row arrangements) : (circular arrangements) equals:ExplanationRow = 5! = 120; circular = 4! = 24. Ratio = 120 : 24 = 5 : 1. Equivalently, 5! / 4! = 5. Hence the answer is 5 : 1.
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Q9 MediumNine persons are to be seated around a circular dining table. How many arrangements are possible?ExplanationArrangements = (9 − 1)! = 8! = 40320. Hence the answer is 40320.
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Q10 MediumFive distinct beads are fixed on a circular bracelet; only rotations are treated as identical (no flipping). How many arrangements exist?Explanation(5 − 1)! = 4! = 24. Hence the answer is 24.
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Q11 HardIn how many ways can 5 boys and 3 girls sit around a circular table so that no two girls are together?ExplanationFirst arrange 5 boys in a circle: (5 − 1)! = 4! = 24. This creates 5 gaps between boys. Place 3 girls in 3 different gaps: 5P3 = 5 × 4 × 3 = 60. Total = 24 × 60 = 1440. Hence the answer is 1440.
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Q12 HardIn an HR round, 10 people sit around a circular table. How many distinct seating arrangements are possible?ExplanationCircular arrangements = (10 − 1)! = 9! = 362880. Hence the answer is 362880.
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Q13 HardIn how many ways can 6 different coloured beads be strung to form a circular necklace if the necklace can be worn either way (flips look the same)?ExplanationCircular arrangements of 6 beads considering rotations only = (6 − 1)! = 120. Allowing flips (reflections identical) divides by 2: 120 / 2 = 60. Hence the answer is 60.
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Q14 HardIn how many ways can 5 men and 4 women sit alternately in a circle?ExplanationArrange 5 men in a circle first: (5 − 1)! = 4! = 24. This creates 5 gaps; place 4 women in 4 distinct gaps so genders alternate (no two women together): 5P4 = 5 × 4 × 3 × 2 = 120. Total = 24 × 120 = 2880. Hence the answer is 2880.
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DS1 Easy
In a placement seating problem, how many ways can n people sit around a round table?
Statement In = 6 and all people are distinct.
Statement IIRotations of the same seating are considered identical.
ExplanationStatement I with the standard circular rule gives (6 − 1)! = 120. Statement II alone does not give n. Statement I alone is sufficient. Answer: A. -
DS2 Easy
Is the number of circular seating arrangements of n people less than 100?
Statement In = 5.
Statement IIn = 6.
ExplanationI: (5 − 1)! = 24 < 100, so yes. II: (6 − 1)! = 120, which is not less than 100. Statement I alone suffices. Answer: A.
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DS3 Medium
In how many ways can 8 people sit around a circular table so that two particular people sit together?
Statement IThe two particular people must sit next to each other.
Statement IIAll 8 people are distinct and rotations are identical.
ExplanationNeed both the together condition and the circular setting: treat the pair as one unit → (7 − 1)! × 2 = 1440. Neither alone fully specifies the count. Answer: C. -
DS4 Hard
How many distinct necklace arrangements are possible with coloured beads?
Statement IThere are 6 distinct beads on a circular necklace.
Statement IIThe necklace can be flipped, so reflections are identical.
ExplanationI alone gives rotations-only count 5! = 120. II alone lacks the number of beads. Together: (6 − 1)! / 2 = 60. Both are needed. Answer: C.