Question

If $m$ parallel lines in a plane are intersected by $n$ parallel lines then number of parallelograms formed is

• A. $\frac{m!n!}{(2!{)}^2}$
• B. $\frac{m!n!}{(m-2)!(n-2)!}$
• C. $\frac{m!n!}{(2!{)}^2(m-2)!(n-2)!}$
• D. $\frac{(m+n)!}{(m+n-2)!2!}$

Answer 1

Answer: (C)

$\frac{m!n!}{(2!{)}^2(m-2)!(n-2)!}$

Each parallelogram is formed by set of two parallel lines. Two parallel lines can be selected from $m$ lines in ${}^m{C}_2$ ways and other two parallel lines can be selected from $n$ lines in ${}^n{C}_2$ ways.

Total number of parallelograms formed is equal to ${}^m{C}_2{\times }^n{C}_2$

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

Hence, the answer is C