Question 8
Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, A_3, \dots, A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \dots, B_n$ lie on $\ell_B$. Additionally, when segments $A_iB_j$ are drawn for all $i = 1, 2, 3, \dots, m$ and $j = 1, 2, 3, \dots, n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m = 7$ and $n = 5$. The figure shows that there are 8 regions when $m = 3$ and $n = 2$.
