Subtopic: Permutations, Combinations & Probability
Question Statement:
How many combinations of non-null sets A, B, C are possible from the subsets of
{2, 3, 5} satisfying the conditions: (i) A is a subset of B, and (ii) B is a subset of C?
Options:
(A) 28
(B) 27
(C) 18
(D) 19
Official Answer Key:
MTA
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.7)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *General Aptitude* | **Topic**: *Quantitative Aptitude* (`Permutations, Combinations & Probability`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **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): `27`**.
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