How many segments go through 2 parallel lines' points?

Question

$2$ parallel lines $a$ and $b$ have points $A_1,A_2,...,A_n\in a$ and $B_1,B_2,...,B_m\in b$ on them. How many intersection points will there be if we will draw segments $A_iB_j$ ($1\leqslant i\leqslant n,1\leqslant j\leqslant m$), with the contition that no $3$ of these segments can intersect in $1$ point?

Answer 1

Hint: Try it with two points on each line. How many ways of drawing the segments lead to an intersection? Now how many ways to select a pair of points from each line?