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 Easy
6 people sit around a round table. How many arrangements?
Explanation(6−1)! = 5! = 120. -
Q2 Medium
5 beads on a circular bracelet (rotation only). Arrangements?
Explanation(5−1)! = 4! = 24. -
Q3 Medium
8 persons around a round table. Arrangements?
Explanation(8−1)! = 7! = 5040. -
Q4 Easy
4 keys on a circular keyring (rotations same). Arrangements?
Explanation(4−1)! = 3! = 6. -
Q5 Medium
7 people around a round table. Arrangements?
Explanation(7−1)! = 6! = 720. -
Q6 Hard
6 distinct beads on a necklace (flip allowed). Arrangements?
Explanation(6−1)!/2 = 120/2 = 60. -
Q7 Easy
3 people around a round table. Arrangements?
Explanation(3−1)! = 2! = 2.
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Q8 Hard
10 people around a circular table. Arrangements?
Explanation(10−1)! = 9! = 362880. -
Q9 Medium
5 people in a row vs around a table. Ratio (row/circle) is:
Explanation5!/4! = 5. -
Q10 Hard
8 beads on bracelet with flip symmetry. Arrangements?
Explanation7!/2 = 5040/2 = 2520. -
Q11 Easy
2 people around a round table. Arrangements?
Explanation(2−1)! = 1! = 1. -
Q12 Medium
9 persons around a round table. Arrangements?
Explanation(9−1)! = 8! = 40320. -
Q13 Easy
Circular arrangements of 6 people equal linear divided by:
Explanation720/120 = 6. -
Q14 Easy
4 distinct flags on a circular pole (rotations same). Arrangements?
Explanation(4−1)! = 6.
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DS1 Easy
How many circular seating arrangements for n people?
Statement In = 6.
Statement IIRotations are considered identical.
ExplanationStatement I alone: (6−1)! = 120. -
DS2 Hard
How many necklace arrangements with flip symmetry?
Statement IThere are 6 distinct beads.
Statement IIRotations and reflections are identical.
ExplanationNeed both: (6−1)!/2 = 60.
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DS3 Medium
How many ways can 8 people sit around a round table?
Statement IAll 8 seats are distinct positions on the circle.
Statement IIRotations count as the same arrangement.
ExplanationEach alone gives (8−1)! = 5040. -
DS4 Easy
Is the number of circular arrangements less than 100?
Statement In = 5 people sit around a table.
Statement IIn = 6 people sit around a table.
ExplanationI: 4! = 24 < 100. II: 5! = 120, not less than 100 alone.