MathDB

Let

\ell_A

and

\ell_B

be two distinct parallel lines. For positive integers

m

and

n

, distinct points

A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak 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,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak 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

. [asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("

\ell_A

",(-2,0),W); label("

\ell_B

",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("

A_1

",A1,S); label("

A_2

",A2,S); label("

A_3

",A3,S); label("

B_1

",B1,N); label("

B_2

",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]

Problem Statement