P, Q, R, S, T, U, V, and W are seated around a circular table.
I. S is seated opposite to W.
II. U is seated at the second place to the right of R.
III. T is seated at the third place to the left of R.
IV. V is a neighbour of S.
Which of the following must be true?
Options:
(A) 0
(B) 6
(C) 12
(D) 16
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2020 Q.7)**
**1. Governing Concept**:
- **Topic**: Logical Reasoning & Circular Seating Arrangement
- Eight persons: P, Q, R, S, T, U, V, W seated in a circle facing inward:
1. $S$ is seated opposite to $W$ (positions 1 and 5).
2. $U$ is second to the right of $R$; $T$ is third to the left of $R$.
3. $V$ is an immediate neighbor of $S$.
- Testing the topological assignments across remaining slots for $P$ and $Q$:
- Regardless of the alternate placements of $P$ and $Q$ in remaining available seats, the geometry dictates that **P cannot be seated opposite to Q**, because the opposite pairs are already consumed by $S-W$ and the fixed offset pairs of $R, T, U, V$.
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Not necessarily true under all valid seating branches.
- **Option (B) is INCORRECT**: Not universally true.
- **Option (C) [P is not seated opposite to Q] is CORRECT**: An invariant constraint satisfied across all admissible circular permutations.
- **Option (D) is INCORRECT**: False in valid configurations.
**3. Final Answer**:
- **Correct Choice**: **C**
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