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

Answer: (A)

Selection of $2$ parallel lines from $m={ }^{m} C_{2}$
Selection of $2$ parallel lines from $n={ }^{n} C_{2}$
Hence, number of parallelograms $={ }^{m} C _{2} \cdot{ }^{n} C _{2}$
$=\frac{1}{4}(m-1)(n-1) m n$