Consider that I have a set of $m$ parallel lines and another set of $n$ parallel lines. Now they are made to intersect forming multiple parallelograms. Now the question is how many unique parallelograms does these $m \times n$ parallel lines form.
2026-04-01 23:07:06.1775084826
Parallelograms formed by $m \times n$ parallel lines
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What forms a parallelogram? Two pairs of parallel sides.
OK. We're done.
Just select two lines from $m$ lines and two from $n$ lines. This type of selection will always result in a unique parallelogram, and obviously, won't leave any.
Therefore number of parallelogram $\displaystyle = \binom m2 \times \binom n2=\frac{nm(n-1)(m-1)}{4}$