Example 3 As shown in Figure $1, a / / b$, line $a$ has ten points $A_{1}, A_{2}$, $\cdots, A_{10}$, and line
$b$ has nine points $B_{1}, B_{2}, \cdots, B_{9}$
nine points. Connecting each point on
$a$ with each point on
$b$ can result in many line segments, given that no three line segments intersect at one point. How many intersection points do these line segments have in total?
(1996, Seoul International Mathematics Invitational Tournament China Training Team Question)
Solution: Take two points each on $a$ and $b$, the four points determine a unique intersection point. Taking two points from $a$ has $10 \times 9 \div 2=45$ methods, taking two points from $b$ has $9 \times 8 \div 2=36$ methods. In total, there are $45 \times 36=1620$ methods.