Question

A parallelogram is cut by two sets of $m$ lines parallel to the sides. The number of parallelograms thus formed is

• A. $\frac{m^2}{4}$
• B. $\frac{(m+1{)}^2}{4}$
• C. $\frac{(m+2{)}^2}{4}$
• D. $\frac{(m+1{)}^2(m+2{)}^2}{4}$

Answer 1

The correct option is D $\frac{(m+1{)}^2(m+2{)}^2}{4}$

,We will have $2m+2$ sets of parallel line ...(due to inclusion of the $2$ initial parallel sides of the parallelogram).
Hence, the number of parallelogram formed by taking two parallel lines from each set, we get,
$=\left({}^{m+2}{C}_2\right)\left({}^{m+2}{C}_2\right)$
$=({}^{m+2}{C}_2{)}^2$
$={\left[\frac{(m+2)(m+1)}{2}\right]}^2$