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Today, we're here to learn how to count quadrilaterals, so sit back, relax, and go grab a cup of coffee if you want to (or tea, if you prefer).

Just in case you don't know, quadrilateral is a polygon that has 4 sides.

Again let's start off with an example:

How many quadrilaterals are there in the 6$$\times$$4 regular grid below?

Of course we could go old school and count them one by one, but we're gonna go the smarter way.

To construct a quadrilateral, we need two horizontal lines, and two vertical lines.

In the 6$$\times$$4 grid above, there are 7 vertical lines and 5 horizontal lines, we choose 2 from each of them.

Hence, the total number of quadrilaterals in a 6$$\times$$4 grid is ${7\choose 2} \times {5\choose 2} =210$

Now, generally, how do we count the number of quadrilaterals in an $$a\times b$$ grid (where $$a$$ is the width of the grid and $$b$$ is the height of the grid)?

Similarly, to construct a quadrilateral, we need two horizontal lines and two vertical lines.

In an $$a\times b$$ grid, there are $$a+1$$ vertical lines and $$b+1$$ horizontal lines, we choose two from each of them.

So, the number of quadrilaterals in an $$a\times b$$ grid is ${a+1\choose 2}\times{b+1\choose 2}=\frac{ab(a+1)(b+1)}{4}$

The number of quadrilaterals in an $$a\times b$$ grid (where $$a$$ is the width of the grid, $$b$$ is the height of the grid) is $\frac{ab(a+1)(b+1)}{4}$

This is one part of Quadrilatorics.

Note by Tan Kenneth
10 months, 4 weeks ago

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