How many rectangles in a 3 by 4 grid?

Question

A grid has 3 rows and 4 columns of unit squares (so 4 horizontal lines and 5 vertical lines). How many rectangles, of all sizes, can be traced along the grid lines? Squares count. Then give the general formula for an m by n grid and check it on a 1 by 1 grid.

Accepted Answer

Sixty. Choosing the sides. Any rectangle in the grid has a top and a bottom, which are two of the 4 horizontal lines (4 choose 2 = 6 ways), and a left and a right, which are two of the 5 vertical lines (5 choose 2 = 10 ways). Every such choice gives exactly one rectangle, so there are 6 × 10 = 60. Counting by size, as a check. Rectangles of width w and height h: (4 minus w + 1)(3 minus h + 1) positions. Summing over w from 1 to 4 and h from 1 to 3: (4 + 3 + 2 + 1) × (3 + 2 + 1) = 10 × 6 = 60. Same answer, and it shows the formula factorises. General formula. An m by n grid of squares has (m +…

How many rectangles in a 3 by 4 grid?

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