
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
161.7k+ views
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.
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.
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