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How many rectangles are there in a chess board?
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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.
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