A school has 100 students distributed among 1st to 10th standards.
Based on this, which one of the following statements is always correct?
Options:
(A) There are at least 10 students who belong to the same standard.
(B) There is at least one student in each standard.
(C) There are at most 10 students in 10th standard.
(D) The total number of students from 1st to 5th standards is at least 50.
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2026 Q.3)**
**1. Governing Concept**:
- **Topic**: Quantitative Aptitude & Pigeonhole Principle
- If $n$ items are placed into $k$ containers, then at least one container must contain at least $\lceil n/k \rceil$ items.
- Here, $n = 100$ students are distributed across $k = 10$ standards (classes).
**2. Option Analysis**:
- The average number of students per standard is $100 / 10 = 10$.
- By the Pigeonhole Principle, at least one standard must contain at least 10 students.
- **Option (A) is CORRECT**: Accurately reflects the mathematical guarantee provided by the generalized Pigeonhole Principle.
- **Options (B), (C), and (D) are INCORRECT**: Assert non-guaranteed strict upper bounds or impossible distributions.
**3. Final Answer**:
- **Correct Choice**: **A**
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