5. Given are two parallel lines $a_{1}$ and $a_{2}$ and $m$ points on $a_{1}$ and $n$ points on $a_{2}$. Each of the $m$ points on line $a_{1}$ is connected to each of the $n$ points on line $a_{2}$. If at no point between the lines $a_{1}$ and $a_{2}$ do more than two of the resulting segments intersect, determine the number of these intersection points.
Solution. All four-element subsets of the set that contain two of the given points from each of the given lines determine exactly one of the described intersection points and vice versa. Therefore, the desired number is equal to $\binom{m}{2}\binom{n}{2}$.