Question Statement:
The coefficient of x4 in the polynomial (x - 1)3(x - 2)3 is equal to _______.
Options:
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(A) 33
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(B) ̶ 3
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(C) 30
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(D) 21
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.9)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *General Aptitude* | **Topic**: *Quantitative Aptitude* (`Numerical Computation & Estimation`)
- **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): `̶ 3`**.
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