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
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.3)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given:
- Total number of students: $N = 100$
- Number of standards (classes): $k = 10$ (from 1st to 10th standard).
- We must determine which statement is **always correct** regardless of how the students are partitioned among the 10 standards.
**2. Step-by-Step Mathematical Derivation**:
1. **Application of the Pigeonhole Principle**:
- By the generalized Pigeonhole Principle, if $N$ items (students) are put into $k$ pigeonholes (standards), then at least one pigeonhole must contain at least:
\left\lceil \frac{N}{k} \right\rceil = \left\lceil \frac{100}{10} \right\rceil = 10 \text{ students}
- Therefore, there must exist at least one standard with $\ge 10$ students. In other words, *"There are at least 10 students who belong to the same standard."*
2. **Evaluation of Other Options by Counterexamples**:
- *Option B*: A standard could have 0 students (e.g., all 100 in 1st standard). False.
- *Option C*: The 10th standard could have all 100 students ($100 > 10$). False.
- *Option D*: All 100 students could be in the 6th to 10th standards, meaning 1st to 5th standards have 0 students ($0 < 50$). False.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) There are at least 10 students who belong to the same standard**: ✅ **CORRECT**. Directly follows from the Pigeonhole Principle $\lceil 100/10 \rceil = 10$.
- **(B) There is at least one student in each standard**: ❌ **INCORRECT**. The problem statement does not require non-empty standards; a standard could have 0 students.
- **(C) There are at most 10 students in 10th standard**: ❌ **INCORRECT**. The 10th standard could easily have 50 or 100 students.
- **(D) The total number of students from 1st to 5th standards is at least 50**: ❌ **INCORRECT**. Distribution could be heavily skewed towards higher standards (e.g., 90 students in 9th and 10th).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: When a question asks for a condition that is "always correct" for arbitrary distributions of discrete objects, immediately invoke the generalized Pigeonhole Principle: $\lceil N/k \rceil$.