# Find the number of triangles given n lines

1. Nov 21, 2012

### nano Math

1. The problem statement, all variables and given/known data
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?

2. Relevant equations
Combination nCr (n choose r)

3. 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 any one please help?

2. Nov 21, 2012

### haruspex

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

3. Nov 21, 2012

### 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 any one please provide insight

4. Nov 21, 2012

### HallsofIvy

Staff Emeritus
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.

5. Nov 21, 2012

### 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

6. Nov 21, 2012

### nano Math

its nC3-mC3

7. Nov 21, 2012

### haruspex

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.

8. Nov 21, 2012

### Boorglar

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

9. Nov 21, 2012

### 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

10. Nov 21, 2012

### 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