4. On parallel lines $a$ and $b$, there are given $m$ and $n$ points, respectively. Each of the given points on line $a$ and each of the given points on line $b$ determine a segment. How many intersection points of such pairs of segments are there that do not lie on lines $a$ or $b$?
4. Let $A_{i}, 1 \leq i \leq m$ be points on line $a$, and $B_{k}, 1 \leq j \leq n$ be points on line $b$. Each pair of points $\left(A_{i}, A_{j}\right), i \neq j$, and each pair $\left(B_{k}, B_{l}\right), k \neq l$, determine one intersection point. If $P$ and $Q$ are the numbers of pairs of observed points on lines $a$ and $b$, respectively, then the number of intersections is equal to $P \cdot Q$.

Let's determine the number of pairs of points on line $a$. The first point of the pair can be chosen in $m$ ways, and then the second in $m-1$ ways. This would give a total of $m(m-1)$ pairs. However, pairs $\left(A_{i}, A_{j}\right)$ and $\left(A_{j}, A_{i}\right)$ are the same, so we need to divide $m(m-1)$ by 2. Therefore, the number of pairs on line $a$ is $\frac{m(m-1)}{2}$.

Similarly, the number of pairs of points on line $b$ is $\frac{n(n-1)}{2}$.

Therefore, the total number of intersections is

$$
\frac{m(m-1)}{2} \cdot \frac{n(n-1)}{2}
$$

![](https://cdn.mathpix.com/cropped/2024_05_30_ad525663fbc4e6951048g-03.jpg?height=238&width=514&top_left_y=1786&top_left_x=822)

## MINISTRY OF EDUCATION AND SPORT OF THE REPUBLIC OF CROATIA

## CROATIAN MATHEMATICAL SOCIETY

## MATHEMATICS