number of rectangles in a chess board?

This is a very standard question. The answer is

⁹C₂ x ⁹C₂

Explanation:

The chess board has 9 vertical lines and 9 horizontal lines.
A rectangle will be formed when we chose any two vertical lines and any two horizontal lines.

The number of ways to chose 2 vertical lines from the given 9 vertical lines is ⁹C₂
Similarly, the number of ways to chose 2 horizontal lines from the given 9 horizontal lines is ⁹C₂

The two methods are independent of each other. Soo to say for each set of vertical lines that we have chosen, there are ⁹C₂ independent ways to pick 2 horizontal lines.

So we multiply both these to get the desired result.

I have tried to explain in great detail. I hope this is clear to you. :)