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How do you find how many rectangles of any size are in a 5 by 8 grid?

I am in a problem solving math class and as part of our take home midterm we were asked to find how many rectangles of any size there are in a 5 by 8 grid. The only problem is, the professor NEVER showed us how to solve this problem.

A 5x8 grid has 6 possible places to choose start and end points of the width, a total of (6*5)/2=15 ways. Similarily, there are 9 possible places to choose start and end points of the height, a total of (9*8)/2=36 ways. Miultiplying height choices with width choices gives a total of 15*36=540 possible rectangles.

The general formula for mxn grid is m*n*(m+1)*(n+1)/4