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Problem:

How many triangles do $m$ lines form if

a) Every two lines intersect and no three lines intersect at one point.

b) There are $n$ lines among $m$ lines that are parallel to each other. No other line is parallel to these $n$ lines, and no other two lines are parallel to each other. Again no three lines intersect at one point.

Thank you.

  • 0
    I made a slight additional correction to the title to bring it in line with the body; I think it all makes sense now.2012-12-16

2 Answers 2

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There are $n$ parallel lines. Each pair of the $m-n$ forms a triangle with each of the $n$. So the triangles are ${m-n \choose 2}n=\frac 12(m-n)(m-n-1)n$

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Take one of these lines. Ever pair of lines intersecting it is forming a triangle-- combination of $m-1$ to $2$.

Do this for every line-- excluding the ones you already worked on. So the second line you are looking at has $m-2$ to $2$ additional triangles formed by itself and the ones intersecting it, ... .

All add up to $\sum_{i=2}^{m-1}i\times(i-1)/2$.

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    This is fine if no lines are parallel.2013-03-19