Question

If $m$ parallel lines in a plane are intersected by a family of $n$ parallel lines, the number of parallelograms than can be formed is

• A. $\frac{1}{4}mn(m-1)(n-1)$
• B. $\frac{1}{2}mn(m-1)(n-1)$
• C. $\frac{1}{4}{m}^2{n}^2$
• D. None of these

Answer 1

Answer: (A)

$\frac{1}{4}mn(m-1)(n-1)$

The number of selection of two parallel lines from m lines is ${}^m{C}_2$.
The number of selection of two parallel lines from n lines is ${}^n{C}_2$.
Hence, the number of parallelograms lines is
${}^m{C}_2{\times }^n{C}_2=\frac{1}{4}mn(m-1)(n-1)$