2009 UNCO Math Contest II Problems/Problem 6

Problem

Let each of $m$ distinct points on the positive $x$ -axis be joined to each of $n$ distinct points on the positive $y$ -axis. Assume no three segments are concurrent (except at the axes). Obtain with proof a formula for the number of interior intersection points. The diagram shows that the answer is $3$ when $m=3$ and $n=2.$

$[asy] draw((0,0)--(0,3),arrow=Arrow()); draw((0,0)--(4,0),arrow=Arrow()); for(int x=0;x<4;++x){ for(int y=0;y<3;++y){ D((x,0)--(0,y),black); }} dot(IP((2,0)--(0,1),(1,0)--(0,2))); dot(IP((3,0)--(0,1),(1,0)--(0,2))); dot(IP((3,0)--(0,1),(2,0)--(0,2))); [/asy]$

Solution

Notice that choosing two points on the x axis and two points on the y axis, then, after constructing all possible lines, there will be only one point of intersection. So the answer is

$\binom{m}{2} \binom{n}{2}$

2009 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
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2009 UNCO Math Contest II Problems/Problem 6

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