A parallelogram is cut by two sets of m lines parallel to its sides.

Question:

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

Solution:

Each set of parallel lines consists of $(m+2)$ lines.

Each parallelogram is formed by choosing two lines from the first set and two straight lines from the second set.

$\therefore$ Total number of parallelograms $={ }^{m+2} C_{2} \times{ }^{m+2} C_{2}=\left({ }^{m+2} C_{2}\right)^{2}$

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