Question 9
Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n,$ distinct points $A_1,$ $A_2,$ $A_3,$ $\ldots,$ $A_m$ lie on $\ell_A,$ and distinct points $B_1,$ $B_2,$ $B_3,$ $\ldots,$ $B_n$ lie on $\ell_B.$ Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i = 1,$ $2,$ $3,$ $\ldots,$ $m$ and $j = 1,$ $2,$ $3,$ $\ldots,$ $n,$ 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 $n = 2.$
