Pariksha SaathiQuantitative Aptitude
Circular Permutations: Practice Questions with Answers
14 practice questions on Circular Permutations, with answers; most also have an explanation or a worked solution. Try each one before you open its answer.
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Q.1
In how many ways can 6 distinct beads be arranged to form a necklace?
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Answer: 60
Solution:
- A necklace can be flipped over, so divide the circular count by 2: (n−1)!/2.
- = (6−1)!/2 = 120/2 = 60.
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Q.2
In how many ways can 4 people be seated around a circular table if two particular people must always sit together?
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Answer: 4
Solution:
- Glue the two people into one block: this leaves 3 entities (the block + 2 other people) to arrange circularly.
- Circular arrangements of 3 entities = (3−1)! = 2.
- The glued pair can sit in 2 internal orders (either person on the left), so total = 2 × 2 = 4.
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Q.3
In how many ways can 5 people be seated around a circular table?
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Answer: 24
Solution:
- Circular arrangements of n people = (n−1)!.
- = (5−1)! = 4! = 24.
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Q.4
In how many ways can 7 people be seated around a circular table if two particular people must never sit together?
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Answer: 480
Solution:
- Total circular arrangements of 7 people = (7−1)! = 720.
- Arrangements with the two together: glue them into a block (6 entities), circular arrangements = (6−1)! = 120, × 2 internal orders = 240.
- Never together = total − together = 720 − 240 = 480.
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Q.5
In how many ways can 4 men and 4 women be seated around a circular table so that men and women alternate?
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Answer: 144
Solution:
- Fix one man's seat to remove rotational duplicates. The remaining 3 men can be arranged in the other 3 "male" seats in 3! ways.
- The 4 women occupy the 4 seats between the men, in 4! ways.
- Total = 3! × 4! = 6 × 24 = 144.
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Q.6
In how many ways can 5 people be seated around a circular table that has 5 numbered seats (each seat is a distinct, labelled position)?
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Answer: 120
Solution:
- Since the seats themselves are distinguishable, rotating the whole group produces a genuinely different arrangement (person X is now in seat 1 instead of seat 2, say).
- So this is NOT the usual circular case — it's equivalent to arranging 5 people in 5 distinct positions: 5! = 120.
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Q.7
In how many ways can a committee of 4 people be chosen from 7 people and then seated around an ordinary (unmarked) circular table?
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Answer: 210
Solution:
- First choose the 4 people: ⁷C₄ = 35.
- Then arrange that group of 4 circularly: (4−1)! = 6.
- Total = 35 × 6 = 210.
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Q.8
In how many ways can 5 people be seated around a circular table if two particular people must always sit together?
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Q.9
In how many ways can 8 people be seated around a circular table if two particular people must never sit together?
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Answer: 3,600. Total (8−1)! = 5,040; together = (7−1)!×2 = 1,440; 5,040 − 1,440.
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Q.10
In how many ways can 5 distinct beads be arranged to form a necklace?
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Q.11
In how many ways can 3 men and 3 women be seated around a circular table so that men and women alternate?
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Q.12
In how many ways can a committee of 3 people be chosen from 6 people and then seated around an ordinary circular table?
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Q.13
In how many ways can 6 people be seated around a circular table that has 6 numbered seats?
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Q.14
In how many ways can 6 people be seated around a circular table?
How these questions were made
Written by Mahesh Srivastava, Founder of Pariksha Saathi. Published: 2026-10-03.