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Rectangles that are not squares

BrainteasersEasy

A board is $5$ cells tall and $8$ cells wide. Count the rectangles, of any size, whose sides lie along the grid lines and that are not squares.

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Hint

Count all rectangles by choosing grid lines, then subtract the squares by side length.

Solution

The board has $6$ horizontal and $9$ vertical grid lines, and a rectangle is fixed by choosing $2$ of each, so there are $(2 6) \times (2 9) = 15 \times 36 = 540$ rectangles in all.
A square of side $k$ has $(6 - k) (9 - k)$ positions, for $k$ from $1$ to $5$ .
That gives $40 + 28 + 18 + 10 + 4 = 100$ squares.
The rectangles that are not squares number $540 - 100 = 440$ .

Answer: $440$