Episode 8 — Aptitude and Reasoning / 8.23 — Arrangements
8.23.a Concepts and Formulas -- Arrangements
1. Types of Arrangements
Type 1: Linear Arrangement
People or objects are arranged in a straight line (row).
Single Row (facing same direction):
Left End Right End
+------+------+------+------+------+
| Pos1 | Pos2 | Pos3 | Pos4 | Pos5 |
+------+------+------+------+------+
(All facing North, say)
Type 2: Circular Arrangement
People sit around a circular table or stand in a circle.
P1
/ \
P5 P2
| |
P4 P3
\ /
--
Type 3: Double Row / Two-Row Arrangement
Two rows of people facing each other.
Row 1 (facing South): A B C D E
| | | | |
Row 2 (facing North): P Q R S T
Type 4: Rectangular / Square Arrangement
People sit around a rectangular table (some face inward, some face outward, or at sides).
+---+---+---+
| P1| P2| P3| (North side, facing South)
+----+---+---+---+----+
| P8 | | P4|
+----+ +----+
| P7 | | P5|
+----+---+---+---+----+
| -- | P6| -- |
+---+---+---+
2. Linear Arrangement -- Key Concepts
Position Numbering
From Left: 1 2 3 4 5 6 7
| | | | | | |
From Right: 7 6 5 4 3 2 1
Relationship: Position from Left + Position from Right = Total + 1
If a person is at position 3 from left in a row of 7: Position from right = 7 - 3 + 1 = 5
"Between" Concept
"X sits between Y and Z" means Y and Z are on either side of X:
... Y X Z ... or ... Z X Y ...
"There are 3 people between X and Y" means:
X _ _ _ Y (3 gaps between them, positions differ by 4)
Formula: If there are n people between X and Y, they are n + 1 positions apart.
"Adjacent / Neighbor" Concept
"X and Y are adjacent" means they sit next to each other:
... X Y ... or ... Y X ...
"X and Y are NOT adjacent" means at least one person sits between them.
"End" Positions
Left End = Position 1 from left
Right End = Position 1 from right (= last position)
"X sits at one end" means X is at Position 1 or Position N.
"Immediate Left / Right"
"Y is to the immediate left of X":
... Y X ...
"Y is to the immediate right of X":
... X Y ...
Finding Total Number of People
If X's position from left = L and from right = R:
Total = L + R - 1
3. Circular Arrangement -- Key Concepts
Key Differences from Linear
| Feature | Linear | Circular |
|---|---|---|
| End positions | Yes (2 ends) | No ends |
| Neighbor count | 1 or 2 | Always 2 |
| Unique arrangements (n people) | n! | (n-1)! |
| Clockwise/Anti-clockwise | N/A | Matters |
Facing Direction in Circular
Center-facing (inward):
P1
/ \
P5 P2 All faces point toward the center
| O | O = center
P4 P3
\ /
In this arrangement:
- P2's right neighbor = P3 (clockwise)
- P2's left neighbor = P1 (anti-clockwise)
Outward-facing:
P1 (faces out/North)
/ \
P5 P2 All faces point AWAY from center
| O |
P4 P3
\ /
In this arrangement:
- P2's right neighbor = P1 (anti-clockwise, since facing outward mirrors the directions)
- P2's left neighbor = P3 (clockwise)
Immediate Left/Right in Circular (Facing Center)
Rule: For people facing CENTER:
- Immediate RIGHT = next person CLOCKWISE
- Immediate LEFT = next person ANTI-CLOCKWISE
For people facing OUTSIDE:
- Immediate RIGHT = next person ANTI-CLOCKWISE
- Immediate LEFT = next person CLOCKWISE
"Between" in Circular
"2 people sit between A and B (clockwise from A)" means:
A
/ \
x (rest of circle)
|
y
\
B
A -> x -> y -> B (clockwise)
4. Constraint Types
Definite Constraints (Fixed Positions)
These pin down exact positions:
- "A sits at the left end."
- "B sits in the middle."
- "C is 3rd from the right."
Relative Constraints (Relational)
These define relationships between people:
- "A sits to the immediate left of B."
- "C and D are adjacent."
- "E sits between F and G."
- "There are 2 people between H and I."
Negative Constraints (Restrictions)
These tell you what CANNOT happen:
- "A does NOT sit at any end."
- "B and C are NOT adjacent."
- "D is NOT to the immediate right of E."
Conditional Constraints
These depend on another condition:
- "If A sits at the left end, B sits at the right end."
- "A sits to the left of B but not necessarily adjacent."
5. The "Fix -> Arrange -> Check" Approach
Step 1: FIX
Start with the most constrained element:
- If someone has a fixed position, place them first.
- If a circular arrangement, fix one person to remove rotational symmetry.
Step 2: ARRANGE
Add remaining people based on constraints:
- Place people with relative constraints next.
- Use negative constraints to eliminate impossible positions.
Step 3: CHECK
Verify all constraints are satisfied:
- Go through EVERY given condition.
- Make sure no constraint is violated.
6. Double Row Arrangement Concepts
Setup
Row 1: A B C D E (Facing South)
| | | | |
Row 2: P Q R S T (Facing North)
"Facing" Concept
- A faces P (they are directly opposite).
- B faces Q, C faces R, etc.
Important:
- "Left" and "Right" depend on the person's facing direction!
- Row 1 (facing South): left = East, right = West (from THEIR perspective)
- Row 2 (facing North): left = West, right = East (from THEIR perspective)
Actually, let's standardize:
Row 1 (facing South):
From Row 1's perspective: LEFT side --> RIGHT side
A B C D E
Row 2 (facing North):
From Row 2's perspective: RIGHT side <-- LEFT side
P Q R S T
Note: Row 1's left end = Row 2's right end (from their respective perspectives)
Key Rule:
In two-row problems, "immediate left" and "immediate right" are ALWAYS from the person's own perspective (based on their facing direction).
7. Key Formulas for Arrangement Counting
| Arrangement Type | Formula | Example (5 people) |
|---|---|---|
| Linear (all) | n! | 5! = 120 |
| Circular | (n-1)! | 4! = 24 |
| Linear with constraint (A at end) | 2 x (n-1)! | 2 x 4! = 48 |
| Circular (fixed position) | (n-1)! | 4! = 24 |
| Two specific people adjacent | 2 x (n-1)! | 2 x 4! = 48 (linear) |
| Two specific people NOT adjacent | n! - 2(n-1)! | 120 - 48 = 72 |
8. Common Question Patterns
Pattern 1: "Who sits at position X?"
Direct position-finding question.
Pattern 2: "How many people sit between A and B?"
Count the positions between two people.
Pattern 3: "Who is to the immediate left/right of A?"
Find the neighbor.
Pattern 4: "Who sits opposite A?" (Circular / Double Row)
Find the person directly across.
Pattern 5: "Which of the following is TRUE?"
Check each option against the arrangement.
Pattern 6: "Which of the following is a possible arrangement?"
Test each given arrangement against all constraints.
9. Dealing with "Left of" vs. "Immediate Left of"
"A is to the left of B" (General)
A is ANYWHERE to the left of B (not necessarily adjacent):
Valid: A _ _ B or A _ B or A B
"A is to the immediate left of B" (Specific)
A is DIRECTLY next to B on B's left:
Only valid: A B
"A is second to the left of B"
There is exactly one person between A and B, with A on the left:
Only valid: A _ B
10. Facing Direction in Linear Arrangement
When "facing North" or "facing South" is specified:
Facing North (observer sees them from the South):
Observer's view:
Left of Observer = Right of the people facing North
Their Left <- P1 P2 P3 P4 P5 -> Their Right
Important: "Immediate left of P3" = P2 (from P3's perspective facing North)