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How many rectangles are there in a chess board?

Visual way to understand: Video link

A rectangle comprises two horizontal lines and two vertical lines. Count the number of pairs of horizontal and number of pairs of vertical lines possible in a chess board to get the answer.

There are $9$ vertical lines and $9$ horizontal lines on the chess board.

To form a rectangle you must choose $2$ of the $9$ vertical lines and $2$ of the $9$ horizontal lines.

For the two horizontal lines: the first line can be chosen in $9$ ways the second in eight ways. This would imply that you could tell the difference between lines $1$ and $3$ say and $3$ and $1,$ which is not the case so you need to divide $9 \times 8$ by $2,$ making $36.$

Similarly you can choose the two vertical lines in $36$ ways.

So the number of rectangles is given by $36 \times 36$

$\Rightarrow 36 \times 36 = 1296$ rectangles.

### 1 comment

We can observe a pattern when taking 1x1 shape, then 2x2 shape and so on, while (manually) counting number of rectanglesin each. So we get for:

1x1: 1 rectangles

2x2: 9 rectangles

3x3: 36 rectangles

Here we observe a pattern that is

$1^2$, $3^2$, $6^2$, $10^2$

which goes like 1,(1+2),(1+2+3),(1+2+3+4) //each term squared

then on continuing this way we get

5x5: $15^2$ rectangles, 6x6: $21^2$ rectangles, 7x7=$28^2$ rectangles

and for 8x8: $36^2$ rectangles= 1296 rectangles

I am not sure if this way is correct.