Question Statement:
S. T. U, V. W. X. Y, and Z are seated around a circular table. T's neighbours are Y and V. Z is
seated third to the left of T and second to the right of S. U's neighbours are S and Y; and T and W
are not seated opposite each other. Who is third to the left of V?
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2017 Q.62)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Analytical Reasoning (Circular Seating Arrangement)
**2. Detailed Step-by-Step Derivation**:
- Eight people ($S, T, U, V, W, X, Y, Z$) sit around a circular table with 8 equally spaced seats (positions 1 to 8 numbered clockwise). All face the center.
- Let us deduce their relative positions systematically:
1. *T's neighbours are Y and V*:
Place $T$ at position 1. Then positions 8 and 2 are occupied by $Y$ and $V$ in some order.
2. *Z is seated third to the left of T*:
Looking inward, "left" is clockwise.
$\text{Position of } Z = 1 + 3 = 4$
3. *Z is seated second to the right of S*:
"Right" is counter-clockwise. If $Z$ at 4 is 2 steps counter-clockwise from $S$, then $S$ must be at position:
$\text{Position of } S = 4 + 2 = 6$
4. *U's neighbours are S and Y*:
$U$ must sit between $S$ (at 6) and $Y$. Since $S$ is at 6, $U$ must be at position 5 or 7.
- But from Clue 1, $Y$ must be adjacent to $T$ (at 8 or 2).
- If $Y$ is at position 8 (adjacent to $T$), then $U$ at position 7 is adjacent to both $Y$ (at 8) and $S$ (at 6)! This works perfectly.
- Therefore: $Y = 8$, $U = 7$, and consequently $V = 2$ (the other neighbour of $T$).
5. Current seat assignments:
$\text{Pos 1: } T, \quad \text{Pos 2: } V, \quad \text{Pos 4: } Z, \quad \text{Pos 6: } S, \quad \text{Pos 7: } U, \quad \text{Pos 8: } Y$
Empty positions remaining: **Pos 3** and **Pos 5**, to be filled by $W$ and $X$.
6. *T and W are not seated opposite each other*:
In an 8-person table, the seat directly opposite position 1 ($T$) is position $1 + 4 = 5$.
Since $W$ cannot be opposite $T$, $W \neq 5$.
Therefore, $W$ must be at **Pos 3**, and $X$ must be at **Pos 5**.
7. Final circular arrangement (clockwise, 1 to 8):
$T (1), \quad V (2), \quad W (3), \quad Z (4), \quad X (5), \quad S (6), \quad U (7), \quad Y (8)$
- We need to find: **Who is third to the left of V?**
- $V$ is at position 2.
- Facing the center, "left" is clockwise:
$\text{Target position} = 2 + 3 = 5$
- At position 5 sits **X**.
- **Option-by-Option Evaluation**:
- **Option (A) [CORRECT]**: X is seated exactly third to the left of V (at position 5).
- **Option (B) [INCORRECT]**: W is at position 3 (first to the left of V).
- **Option (C) [INCORRECT]**: U is at position 7.
- **Option (D) [INCORRECT]**: T is at position 1 (first to the right of V).
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **A**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).