IIT JEE , AIEEE Forum - Mathematics , Algebra

prahalad

In a plane there are n lines , no two are parallel and no three are concurrent. How many triangles can be formed with their point of intersection as vertices?

cooldude

total no of points is nC2=n(n-1)/2.
for triangles select any 3 of them.that can be done in
[n(n-1)/2]C3 ways.

prahalad

plz explain briefly

thevyzz

since none of em are parallel and concurrent , all of them instersect with one another in an specific pt( ie three lines have their pt of intersection same)

twice total no of intersections made = n(n-1)

no of intersections = n(n-1)/2

no of vertices available = n(n-1)/2

since three pts are req for a triangle,

no of triangles formed = [n(n-1)/2]C3 = [(n(n-1)/2)!] / [6*( n(n-1)/2 -- 3)! ]

rohith291991

ok.. when there are 2 lines there is one intersection when there are 3 lines there are 3.. when there are four there are 6...etc.. this is a series where the common difference is in ap.. and the total number of ntersections for n lines comes out to be n(n-1)/2 out of which n-1 are collinear therefore the number of triangles that can be formed is n(n-1)/2C₃ -(n-1)C₃.... will post full working soon if right plz rate....

Rishi_08

Since there are n lines so that no. of their pt. of intersection=nC2=n(n-1)/2

But ther are n-1 pts on each line. These n-1 collinear pts are actually the pts of intersection of the line with other n-1 lines.

Now we can select there pts from each set of n-1 pts in (n-1)C3 ways

But as these n-1 pts are collinear & hence will n't give any triangle. In this way n. (n-1)C3 combinations give no triangle.

Therefore reqd no of triangles=(1/48)n(n-1)(n^4-2n^3-13n^2+46n-40)

Hope it will help u

cooldude

since there is no parallel line total no of points of intersection is same as selecting 2 lines from n lines.for every 2 lines we have a point of intersection.it can be done in nC2 ways.
for a triangle we require 3 points.for every 3 points we can draw a triangle.selecting 3 points from nC2 lines can be done in (nC2)C3 ways.

prathima

is the answer n(n-1)/2 C3 - n(n-2)C3??????????
plzzzzzzz send the answer if it is wrong

nano0101

yes quite tricky it is !

prahalad

The soln provided by rishi is quite useful and ans given by him is right.

Thanks Mr Rishi

Ask & Discuss Questions with Community & Experts