Find the number of triangles given n lines

[nano Math]

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?

 

[haruspex]

In #2? Which triples of lines will form a triangle? Not just any triple, right?

 

[nano Math]

i am assuming it might be of form

but i see m*(nC3) doesn't work here. here there are 4 triangles. can anyone please provide insight

 

[HallsofIvy]

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.

 

[nano Math]

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

 

[nano Math]

its nC3-mC3

 

[haruspex]

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.

 

[Boorglar]

Perhaps you might want to consider triangles formed from two special sets of lines.

 

[Mouse07]

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

 

[Mouse07]

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