How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a 5*5 square array of dots, as in the figure below?
(Two rectangles are different if they do not share all four vertices.)
A rectangle can be made with 2 vertical dots and 2 horizontals.
For the 2 vertical dots there are 5 to choose from.
that is choosing 2 dots from 5 or 5 choose 2.
Now applying the same logic, we also need to choose 2 horizontal dots.
There are 5 dots to choose from. so we have 5 choose 2
5 choose 2 is 10. there are 10 choices for horizontal and 10 choices for vertical. 10x10=100 choices for a rectangle