Question

If m parallel lines in a plane are intersected by a family of n parallel lines, calculate the number of parallelograms formed.

• A. mn
• B.
• C. [mn (m-1)(n-1)]
• D. [mn (m+1)(n+1)]
• E.

Answer 1

The correct option is C [mn (m-1)(n-1)]

A parallelogram is formed when two select 2 lines from the group of m lines, and 2 from the group of n line. This can be done in =${}^m{C}_2$and${}^n{C}_2$ways. So, total no. of parallelograms formed${=}^m{C}_2{\times }^n{C}_2=\left(\frac{1}{4}\right)\left[\frac{(m!n!)}{\left(\right(m-2)!(n-2)!)}\right]=\frac{1}{4}\left[mn\right(m-1\left)\right(n-1\left)\right]$

Hence option (c)

2nd method:By substitution of any values in choices. Let $m=2\&n=2$ then total parallelograms =1. Only option C will satisfy. Then option (c).