Q.7 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

Introduction

When two sets of parallel lines intersect, they form a grid-like pattern that can generate many parallelograms of different sizes. Understanding how to count these parallelograms is a common and important concept in combinatorics and geometry-based aptitude questions, especially in competitive exams. This problem uses combinations to systematically count all possible parallelograms without missing any.

Goal: Find the correct option and understand the reasoning in detail.

Core Concept: How Parallelograms Are Formed

To form a parallelogram from two families of parallel lines:

• One set has 4 parallel lines (say horizontal).
• The other set has 5 parallel lines (say slant or vertical-like but parallel among themselves).

A parallelogram is completely determined by:

• Choosing 2 lines from the first set (these form one pair of opposite sides).
• Choosing 2 lines from the second set (these form the other pair of opposite sides).

Therefore:

Number of ways to choose 2 lines from 4 lines = (4C2).

Number of ways to choose 2 lines from 5 lines = (5C2).

Total number of parallelograms = (4C2) × (5C2)

Now compute:

(4C2) = 4! / (2! × 2!) = (4 × 3) / (2 × 1) = 6

(5C2) = 5! / (2! × 3!) = (5 × 4) / (2 × 1) = 10

So:
Total Parallelograms = 6 × 10 = 60

So the correct answer is 60, which corresponds to option (C).

Detailed Option-wise Explanation

Option (A) 20

If someone gets 20, they likely:

• Use 4 × 5 = 20 (smallest regions only)
• Ignore larger parallelograms formed by skipping lines

Therefore 20 is too small.

Option (B) 48

48 appears due to errors such as:

• Using only one correct combination
• Incorrectly subtracting overlapping counts

Since 4C2 = 6 and 5C2 = 10, any answer other than 6 × 10 = 60 is wrong.

Option (C) 60 (Correct)

Correct reasoning:

• Choose 2 of 4 lines → 6 ways
• Choose 2 of 5 lines → 10 ways

Each selection forms one parallelogram.

Total = 6 × 10 = 60 parallelograms

Option (D) 72

72 suggests overcounting or formula misuse. It exceeds the combinatorial result, so it cannot be correct.

General Formula and How to Remember It

For two sets of parallel lines:

Total parallelograms = mC2 × nC2

Here:

• m = 4
• n = 5

Total = 4C2 × 5C2 = 6 × 10 = 60

This method works for any similar MCQ involving two families of parallel lines in plane geometry and is very useful for aptitude and competitive exams.