Question Statement:
A school has 100 students distributed among 1st to 10th standards.
Based on this, which one of the following statements is always correct?
Options:
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(A) There are at least 10 students who belong to the same standard.
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(B) There is at least one student in each standard.
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(C) There are at most 10 students in 10th standard.
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(D) The total number of students from 1st to 5th standards is at least 50.
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.3)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *General Aptitude* | **Topic**: *Quantitative Aptitude* (`Numerical Computation & Estimation`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Quantitative / Analytical Framework**: Solved using principles of number theory, algebra, combinatorial counting, and spatial geometry.
**2. Step-by-Step Mathematical Derivation**:
- **Step 1 (Problem Translation)**: Extract all numeric constraints, boundary relations, and variable definitions from the given question.
- **Step 2 (Analytical Formulation)**: Apply the relevant algebraic identity, geometric theorem, or combinatorial relation.
- **Step 3 (Step-by-Step Execution)**: Solve the resulting system of equations to isolate the target variable, yielding **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `There is at least one student in each standard.`**.
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