1. ## Parallels and Parallelograms

Hey guys, I'm having some trouble with this problem.

The objective of this is to create an equation that would determine how many parallelograms would be created through "m" horizontal parallel lines intersected by "n" parallel transversals.

The first problem is to determine an equation for only two parallel lines intersected by "x" number of transversal lines.

_____/____/_____

___ /___ /______

So thats how it looks like... when two parallel lines are intersected by two transversal lines, it creates one parallelogram.

Then, three transversal lines are added to form three parallelograms, the two by itself and the two combine to make a bigger.
_____/____/___/__

____/___ /___/__

This can be done further with 5,6,7 transversals. How can I get a equation from this? If you do get an equation, I need to know HOW exactly the equation is formed (very important) and I need to show tables and patterns and such.

Then, the next problem is to determine the number of parallelograms formed by three horizontal parallel lines intersected by "x" number of parallel transversals. I need to generate a formula for that too, and HOW I made the formula needs to be stated.

After all this, I have to find the amount of how many parallelograms are formed by "m" horizontal parallel lines intersected by "n" parallel transversals. I need to find an equation for this too... and again, I need to state how I came/made this formula.

2. There are $m\choose2$ ways of choosing the two horizontal lines, and $n\choose2$ ways of choosing the two sloping lines. So the total number of parallelograms that can be formed is ${m\choose2}{n\choose2} = \tfrac14m(m-1)n(n-1)$.