A set of 4 parallel lines intersect with another set of 5 parallel lines. How many
parallelograms are formed?
(A) 20 (B) 48 (C) 60 (D) 72
Options:
(A) 20
(B) 48
(C) 60
(D) 72
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2018 Q.07)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Combinatorics (Forming Parallelograms from Intersecting Parallel Lines)
**2. Detailed Step-by-Step Derivation**:
- A parallelogram is formed by choosing two lines from one set of parallel lines and two lines from an intersecting set of parallel lines.
- Number of lines in the first parallel set = $m = 4$.
- Number of ways to choose 2 lines from the first set:
$inom{4}{2} = rac{4 imes 3}{2 imes 1} = 6$
- Number of lines in the second parallel set = $n = 5$.
- Number of ways to choose 2 lines from the second set:
$inom{5}{2} = rac{5 imes 4}{2 imes 1} = 10$
- By the fundamental principle of counting, total number of distinct parallelograms formed:
$N = inom{4}{2} imes inom{5}{2} = 6 imes 10 = 60$
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: 20 equals $4 imes 5$, which is the number of intersection points, not parallelograms.
- **Option (B) [INCORRECT]**: 48 is an incorrect combinatorial product.
- **Option (C) [CORRECT]**: 60 is the exact number of parallelograms formed ($inom{4}{2} imes inom{5}{2} = 60$).
- **Option (D) [INCORRECT]**: 72 is an erroneous combinatorial estimate.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **C**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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