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Circular Permutations Questions with Solutions

Arranging people around a circle is different from arranging them in a row — rotating everyone one seat over gives the "same" arrangement, so one fixed reference point gets removed from the count. Here’s the concept, four fully worked examples, and a few to try yourself.

The concept, quickly

  • n distinct people around a circular table: (n−1)! arrangements, not n! — fixing one person’s seat removes the rotational duplicates (rotating everyone by one seat is still counted as the same arrangement).
  • For a necklace or bracelet (where flipping it over gives the same arrangement, not just rotating it), divide by 2 as well: (n−1)!/2.
  • When two specific people must sit together, glue them into a single block first. Arrange the resulting (reduced count) of entities in a circle using (count−1)!, then multiply by 2 for the two ways the glued pair can face each other.
  • For "must NOT sit together," find the total circular arrangements, subtract the "sit together" count — this is faster and less error-prone than trying to count the "not together" cases directly.

Worked examples

1. In how many ways can 5 people be seated around a circular table?

Solution

  1. Circular arrangements of n people = (n−1)!.
  2. = (5−1)! = 4! = 24.

Answer: 24

2. In how many ways can 6 distinct beads be arranged to form a necklace?

Solution

  1. A necklace can be flipped over, so divide the circular count by 2: (n−1)!/2.
  2. = (6−1)!/2 = 120/2 = 60.

Answer: 60

3. In how many ways can 4 people be seated around a circular table if two particular people must always sit together?

Solution

  1. Glue the two people into one block: this leaves 3 entities (the block + 2 other people) to arrange circularly.
  2. Circular arrangements of 3 entities = (3−1)! = 2.
  3. The glued pair can sit in 2 internal orders (either person on the left), so total = 2 × 2 = 4.

Answer: 4

4. In how many ways can 7 people be seated around a circular table if two particular people must never sit together?

Solution

  1. Total circular arrangements of 7 people = (7−1)! = 720.
  2. Arrangements with the two together: glue them into a block (6 entities), circular arrangements = (6−1)! = 120, × 2 internal orders = 240.
  3. Never together = total − together = 720 − 240 = 480.

Answer: 480

Try these yourself

Use (n−1)! as the base, adjusting for a necklace or a together/not-together condition as needed, then check against the answer.

1. In how many ways can 6 people be seated around a circular table?

Answer: 120

2. In how many ways can 5 distinct beads be arranged to form a necklace?

Answer: 12

3. In how many ways can 5 people be seated around a circular table if two particular people must always sit together?

Answer: 12

Where this comes up

Circular Permutations is a regular sub-topic within the Quantitative/Numerical Aptitude section of every exam Pariksha Saathi covers — SSC CGL, SSC MTS, SSC CHSL, IBPS PO, IBPS Clerk, SBI PO and SBI Clerk — building directly on Permutation and Combination.

Practice more Circular Permutations questions

This page covers the concept and a handful of worked examples. Pariksha Saathi has full topic-wise practice sets for Circular Permutations (and every other Quant/Reasoning/English topic) with instant scoring and explanations — free, no account required.

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