Q68 / 200 · Manipal (MET) 2013 · Mathematics

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

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

Solution

Selection of 2 parallel lines from $m={ }^m \mathrm{C}_2$ Selection of 2 parallel lines from $n={ }^n C_2$ Hence, number of parallelograms $={ }^m \mathrm{C}_2 \cdot{ }^n \mathrm{C}_2$ $=\frac{1}{4}(m-1)(n-1) m n$

Hence, the correct answer is option: 'A'.

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