I2.3 In a $(2 b+1) \times(2 b+1)$ checkerboard, two squares not lying in the same row are randomly chosen. If $c$ is the number of combinations of different pairs of squares chosen, find the value of $c$.
There are 25 squares. First we count the number of ways of choosing two squares lying in the same column or the same row: ${ }_{5} C_{2} \times 5+{ }_{5} C_{2} \times 5=100$
$$
\therefore c={ }_{25} C_{2}-100=200
$$

Method 2 Label the two squares as $A, B$.
For each chosen square $A$ (out of 25 squares), $B$ has 16 possible positions.
$\therefore$ There are $25 \times 16=400$ combinations.
However, $A, B$ may be inter-changed. $\therefore$ We have double counted. $c=200$.