2020/006) How many squares and rectangles are there on a standard chess board

First let us calculate the number of squares


For the square of side n we can choose in rows in 9-n ways(side 1 8 ways, side 2
7 ways son on). In the column in 9-n ways. so the number of ways the
square of side n can be chosen in $(9-n)^2$ ways and as we have number of sides from 1 to 8
so number of ways = $$\sum_{k=1}^{8}(9-k)^2= \sum_{n=1}^{8}(n)^2 = 204$$

Now for calculation of number of rectangles.


For a rectangle we need to choose 2 lines in rows $9 \choose 2$ ways and in
columns in $9 \choose 2$ ways so total number of ways ${9 \choose 2  }^2$ or
$(\frac{9 * 8}{2})^2$ ways that is 1296 ways. As there are 204 squares
so number of rectangles = 1296-204 = 1092