On an infinite checkerboard, the union of any two distinct unit squares is called a (disconnected) domino. A domino is said to be of type $(a, b)$, with $a \leq b$ integers not both zero, if the centers of the two squares are separated by a distance of $a$ in one orthogonal direction and $b$ in the other. (For instance, an ordinary connected domino is of type ( 0,1 ), and a domino of type $(1,2)$ contains two squares separated by a knight's move.)
Each of the three pairs of squares above forms a domino of type $(1,2)$.

Two dominoes are said to be congruent if they are of the same type. A rectangle is said to be $(a, b)$-tileable if it can be partitioned into dominoes of type $(a, b)$.
1. [15] Let $0<m \leq n$ be integers. How many different (i.e., noncongruent) dominoes can be formed by choosing two squares of an $m \times n$ array?
Solution: We must have $0 \leq a<m, 0 \leq b<n, a \leq b$, and $a$ and $b$ not both 0 . The number of pairs $(a, b)$ with $b<a<m$ is $m(m-1) / 2$, so the answer is
$$
m n-\frac{m(m-1)}{2}-1=m n-\frac{m^{2}-m+2}{2} .
$$