50 Word Summary

Circular arrangements are a permutation case where only relative positions matter. This reflects rotational equivalence: rotating does not change order. The count is (n−1)! because one position must be fixed. Example: seating four friends around a round table yields (4−1)! = 3! = 6 arrangements.

 

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Overview

Circular arrangements form a unique branch of permutation problems because relative positions, not absolute ones, define the outcome. Unlike linear seating, rotations of the entire group do not create new arrangements, which is why the formula changes to (n − 1)!. In the explanatory video for this article, followed by the discussion here, we explore several variations that build upon this principle, including groups seated together, restrictions on who may or may not sit beside one another, and the impact of numbered seats. Understanding these cases is an essential part of effective GMAT prep, since such problems test the ability to adapt core concepts to new structures while maintaining logical clarity and accuracy.

 

Circular Arrangements Explained

When people are seated in a circular arrangement, the total number of distinct arrangements is not the same as in a row.

For example, if six people sit in a circle, shifting all of them by one, two, or more seats does not change their neighbours. The arrangement is still identical.

To handle this, we fix one person in a seat and arrange the rest around them.

That gives us (n – 1)! arrangements for n people.

So, for 6 people, the total number of ways is 5!.

An Explanatory Example of Circular Arrangements

Question: In how many ways can 10 people be seated along a circular table?

In a line, 10 people can be arranged in 10! ways.

Around a circle, rotations are identical, so we fix one person and arrange the rest.

Number of arrangements = (10 − 1)! = 9!

Therefore, the answer is 9!

Twists

1. What if three particular people of the 10 always sit together?
2. What if two of the ten never sit together?
3. What if these 10 are five couples and each couple sits together?
4. What if the chairs are numbered?

Special Cases in Circular Arrangements

1. Three particular people sit together

   Treat the three as a single unit.

Now there are 8 entities (7 singles + 1 group).

These can be arranged in 7! ways, and the group itself can be arranged in 3! ways.

Therefore, the answer is 7! x 3!

2. Two particular people never sit together

   First, calculate the total (9!).

Then subtract the cases where these two are seated together.

Two particular people sit together in 8! x 2! ways

Therefore, the answer is 9! – 8! x 2!

3. Five couples sit together

   Each couple is one unit, giving 5 entities.

Arrangements in a circle are (5 – 1)! = 4!.

Within each couple, there are 2! ways of seating.

For 5 couples, that becomes (2!)⁵.

Therefore, the answer is 4! × (2!)⁵

      4. Chairs are numbered

If chairs are numbered, the arrangement works like a row, since rotating positions now changes the arrangement.

Therefore, the answer is 10!

Key Takeaways

Circular arrangements highlight how relative positions change the way we count possibilities. Unlike linear seating, rotations do not create new outcomes, which is why the formula reduces to (n − 1)!. Variations, such as groups sitting together, restrictions on who may sit apart, or numbered seats, build further nuance. The core learning is to fix one position and then apply logical adjustments for any added condition. Developing fluency in these problems strengthens both speed and accuracy. Regularly practicing such variations through timed exercises and adequate GMAT mocks ensure that the concept becomes intuitive under exam conditions.

Parting Thought

Circular arrangements remind us that perspective often matters as much as action. In GMAT preparation, learning to fix one element and view the problem from a stable point simplifies complexity, much like anchoring a seat in a circle. In the MBA admissions journey, too, clarity comes when we fix our priorities and allow other tasks to align around them. Life follows the same principle: rotating endlessly without a point of reference leads to confusion, while anchoring values and goals gives meaning to change. Success lies in finding that anchor and building around it with patience and discipline.