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There is a set of parallel lines with $x$ lines in it and another set of parallel lines with $y$ lines in it. The lines intersect at $12$ points. If $x>y$, find the maximum number of parallelograms that can be formed.

My solution approach :-
As there are $12$ points which we are getting after the intersection of $x$ parallel lines with the $y$ parallel lines, then it should mean that $xy=12$. Please confirm me on this if I am correct or not?

Now I solved this question diagrammatically i.e. taking $12$ points formed by intersection of some $x$ parallel lines with the $y$ parallel lines and then counting all the possible parallelograms in the diagram and I counted a maximum of $18$ parallelograms that can be formed in this condition and my answer is correct !

But I was thinking how this question can be solve mathematically, without doing any manual counting for some bigger numbers of intersection points; let's say there are $50$ intersection points or $1000$ intersection points? Also is there any importance of $x>y$? How it would have impacted the answer if the relation would have been $y>x$? Is there a way to solve this question mathematically in bigger number scenarios? Please clarify me on this !!!

Thanks in advance !!!

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  • $\begingroup$ From a solution-verification POV, you have not included enough details. The numerical answer is correct (and if that's all you're going for, great), but the proof part leaves a lot to be desired. EG How do you know that you have considered all possible arrangement of 12 points (esp since that wasn't referenced explicitly). $\endgroup$
    – Calvin Lin
    Commented Jun 15, 2021 at 21:51

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Given: The lines intersect at $12$ points

Therefore $x \cdot y = 12$. Since $x > y$ possibilities are $x = 6$ and $y = 2$ or $x = 4$ and $y = 3$.

We need $4$ points to form a parallelogram, $2$ points forming $1$ line and the other $2$ points forming the other line. When $x = 6$ and $y = 2$, the number of parallelograms formed $= {}_6C_2 \cdot _2C_2 = 15 \cdot 1$

When $x = 4$ and $y = 3$, the number of parallelograms formed $= _4C_2 \cdot _3C_2 $

Compute this and get the answer

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