How many rectangles on the board

Count every rectangle on the board, of any size and shape. A rectangle is fixed by choosing two of the $9$ vertical grid lines for its sides and two of the $9$ horizontal lines for its top and bottom. So the number of rectangles is $\binom{9}{2} \times \binom{9}{2}$.

Compare with the count of squares from the previous module: squares are the special rectangles whose width equals their height. Turning "count the rectangles" into "choose two lines each way" is the key move: a hard enumeration becomes two easy choices multiplied.