Find the number of triangles given n lines

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To determine the number of triangles formed by n non-parallel lines where no three intersect at a point, the solution is given by the combination formula nC3. When considering n lines with m parallel lines, the correct approach involves calculating the triangles formed by the non-parallel lines, which is (n-m)C3, and then adding the contributions from the parallel lines. Each parallel line requires two non-parallel lines to form a triangle, leading to a total triangle count of (n-m)C3 plus contributions from the parallel lines. The discussion emphasizes the need to carefully account for the conditions set by the problem, particularly regarding parallelism. Understanding these relationships is crucial for accurately solving the problem.
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Homework Statement


1. Given n non-parallel lines such that no three intersect in a point, determine how many triangles are formed?

2. Given n lines in total, of which m are parallel, how many triangles are formed?

Homework Equations


Combination nCr (n choose r)


The Attempt at a Solution


for #1, its nC3, since the question is more like how many ways can we choose three lines from n lines.

for #2, i am not sure.
I assume since each parallel line adds a triangle, it would be m*(nC3). we would have m more triangles. I am not sure if this is correct, could anyone please help?
 
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In #2? Which triples of lines will form a triangle? Not just any triple, right?
 
i am assuming it might be of form
2qicxs4.png


but i see m*(nC3) doesn't work here. here there are 4 triangles. can anyone please provide insight
 
Your picture shows four parallel lines and you mention parallel lines but the condition in the statement of the problem is that none of the lines are parallel.
 
the picture is related to question #2, which states of n lines, there are m parallel lines.
assuming n is composed of (n-m) non parallel lines and m parallel lines
 
its nC3-mC3
 
nano Math said:
its nC3-mC3
No, that counts all triples from the n, then removes those where all three were in the m. But if any two are in m it won't form a triangle.
 
Perhaps you might want to consider triangles formed from two special sets of lines.
 
LOL CSCA67 i don't get it either, i put m(h+1 chose 3) where h are line and m-n >= 3 just get something :P
 
  • #10
lol i found the answer it like

n-m are non parallel lines n-m chose 3 that's how many triangle will be formed by non parallel lines
and for each parallel line u need 2 non parallel lines to form a triangle u do that for all the parallel lines
 
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