Hello again fellow quadrilateral enthusiasts!

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 \(6\times4\) 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 \(6\times4\) 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 \(6\times4\) 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

widthof the grid, \(b\) is theheightof the grid) is \[\frac{ab(a+1)(b+1)}{4}\]

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A complex problem has been beautifully simplified. Thanks a lot.

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