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Circular Seating Arrangement

AFCAT · Logical Reasoning

Also prepare for:NDA CDS Agniveer Agni Army

← Linear Seating Arrangement Floor-Based Puzzle →

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Core Concept

Read this first — the foundation of the topic

→Core Concept

In circular seating, positions are relative to each other. There is no first or last position. The key is to fix one person's position first, then arrange others relative to that person

💡Key Rules

1) Always fix one person's position to eliminate rotational possibilities 2) Clockwise and anti-clockwise directions matter 3) 'Opposite' means diametrically across 4) Adjacent means immediate left or right 5) Use process of elimination systematically.

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Formula Block

Memorise — at least one formula appears in every paper

Total circular arrangements of n people = (n-1)! This is because we fix one position to avoid counting rotations as different arrangements. For restrictions like couples sitting together, treat them as single units.

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Exam Patterns

What examiners ask — read before attempting PYQs

→Common question types include

immediate neighbors, opposite positions, counting positions between people, and conditional arrangements. Most questions give 6-8 people with 4-6 conditions

⚡Shortcut Trick 1 - The Fixed Position Method

Always start by placing one person at the 'top' of the circle. This eliminates confusion about rotational arrangements. Mark positions as 1,2,3... clockwise from this fixed person

⚡Shortcut Trick 2 - The Opposite Formula

In a circle of n people, if person A is at position x, then the person opposite to A is at position x + n/2 (if n is even). For odd numbers, no one sits exactly opposite

✏️Worked Example 1

Six friends A,B,C,D,E,F sit around a circular table. B sits two places to the right of A. C sits opposite to A. D sits immediately left of C.

Where does E sit

→Solution Step-by-Step

1

Fix A at position 1 (top)

2

B sits two places right of A, so B at position 3

3

C sits opposite A. In 6-person circle, opposite of position 1 is position 4. So C at position 4

4

D sits immediately left of C. Left of position 4 is position 3. But B is already there. This means left of C is position 5. Wait - let me recalculate. If we number 1,2,3,4,5,6 clockwise, then left of 4 is position 3. Since B occupies 3, there's a contradiction

🔑Let me recheck

immediate left of position 4 is position 5 (going anti-clockwise). So D at position 5. Step 5: Remaining positions are 2 and 6. E and F occupy these

→Answer

E sits at either position 2 or 6

✏️Worked Example 2

Seven people sit in a circle. P sits third to the right of Q. R sits second to the left of P. How many people sit between Q and R when counted clockwise from Q

→Solution Step-by-Step

1

Fix Q at position 1

2

P sits third to right of Q, so P at position 4

3

R sits second to left of P

→Left of position 4 going anti-clockwise

position 3 is first left, position 2 is second left. So R at position 2 Step 4: Count from Q(position 1) to R(position 2) clockwise: only position 7 comes between them Answer: 1 person sits between Q and R

⚡Shortcut Trick 3 - The Gap Counting Formula

To count people between positions A and B clockwise: if B > A, count = B - A - 1. If B < A, count = n - A + B - 1 (where n is total people)

→Most Common Trap

Students often confuse 'left' and 'right' directions

💡Remember

in clockwise numbering, right means higher numbers, left means lower numbers. Also, 'second to the right' means there's one person in between, not sitting in the second chair. Many students miss this and count wrong positions.

🔑 Key Points

Always fix one person's position first to avoid rotational confusion
In circular arrangement of n people, total arrangements = (n-1)!
Opposite position formula: if person at position x, opposite at x + n/2 (for even n)
Second to the right means one person sits in between, not the second position
Clockwise numbering: right = higher numbers, left = lower numbers
Gap counting formula: between A and B clockwise = B-A-1 (if B>A)
For odd number of people, no one sits exactly opposite to anyone
Adjacent means immediate left or immediate right neighbor only
Use elimination method when multiple conditions are given
Draw the circle and number positions 1,2,3... for visual clarity

📌 Exam Facts

Circular arrangements of n distinct objects = (n-1)! permutations
In 6-person circle, opposite positions are (1,4), (2,5), (3,6)
In 8-person circle, positions 1 and 5 are diametrically opposite
SSC CGL typically gives 6-8 people in circular seating questions
Maximum people between any two positions in n-person circle = n-3
In odd-numbered circles (5,7,9 people), no exact opposite positions exist
Most SSC circular arrangement sets contain 3-5 questions with 4-6 conditions
Immediate left of position 1 in n-person circle is position n

🚀 60-Second Revision

Remember: Fix one person first, then arrange others relative to that position
Formula: Opposite position = current position + n/2 (for even n only)
Trap: 'Second to right' means one person between, not second position
Trick: Number positions 1,2,3... clockwise for systematic solving
Direction: Right = clockwise = higher numbers, Left = anti-clockwise = lower numbers
Gap formula: People between A and B = |B-A|-1 positions
Quick check: Total positioned people should equal given number

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