Question

If $m$ parallel lines in a plane are intersected by a family of $n$ parallel lines, then the number of parallelograms that 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

The correct option is A $\frac{1}{4}mn(m-1)(n-1)$

Selection of $2$ parallel lines from $m={}^m{C}_2$

Selection of $2$ parallel lines from $n={}^n{C}_2$

Hence, the number of parallelograms ${=}^m{C}_2{\cdot }^n{C}_2$

$=\frac{m!}{(m-2)!2!}\times \frac{n!}{(n-2)!.2!}$

$=\frac{m(m-1)(m-2)!}{(m-2)!.2}\times \frac{n(n-1)(n-2)!}{(n-2)!.2}$

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