Question

There is a set of $m$ parallel lines intersecting a set of other $n$ parallel lines in a plane. Then find the number of parallelograms formed.
A. ${}^{m-1}{C_2}^{n-1}{C}_2$
B. ${}^m{C_2}^n{C}_2$
C. ${}^{m-1}{C_2}^n{C}_2$
D. ${}^m{C_2}^{n-1}{C}_2$

Answer 1

Hint: Use the properties of the parallelogram and find the number of ways that the pair of parallel lines is taken from each set. In the end, use a number of ways to reach the required answer.

Formula Used:
The combination is a way of selecting objects from a set or collection of objects where an order of selection does not matter.
The number of combinations of $r$ objects chosen from $n$ objects is denoted by: ${}^n{C}_r$

Complete step by step solution:
Given: A set of $m$ parallel lines intersects with a set of another $n$ parallel lines in a plane.
We know that the opposite sides of a parallelogram are parallel.
So, the parallelogram is formed by the two pairs of parallel lines.

Since there are 2 sets of parallel lines.
So, either we choose the parallel lines from the first set or from the second set.
Case 1: When we choose the lines from a set of $m$ parallel lines
The number of ways of taking the lines: ${}^m{C}_2$

Case 2: When we choose the lines from a set of $n$ parallel lines
The number of ways of taking the lines: ${}^n{C}_2$

Therefore, the total number of parallelograms formed is:
Total parallelogram $={}^m{C_2}^n{C}_2$

Option ‘B’ is correct

Note: Students often make mistakes while calculating the total number of parallelograms. Sometimes they add the number of ways instead of multiplying it. So make sure you multiply the terms.