Q)triangle has n dots on each side. (as shown in fig) How many triangles can be formed (any size).

1st image (image002.jpg)

My incomplete solution

Ans)

If thr are n dots then total no of dots =n(n+1)/2

No of combination using 3 at a time =n(n+1)/2C3

but all set of 3 pts wont be ∆

some will be co linear points

image 2 (1111.jpg)

likex1,x2,x3are coliner ON lineAB, similarly onCD,EFn so on

n similarly on other side of ∆BT, GHn so on &AT ,SJn so on

combination of 3pts on AB =nC3

on CD =(n-1)C3

on EF =(n-2)C3

so total =nC3 +(n-1)C3+(n-2)C3………………………………….. 3C3

as there are 3 side therefor:

3x(nC3 +(n-1)C3+(n-2)C3………………………………….. 3C3)= 3(n∑ i=3 iC3)(the limits of summation didnt come properly)

so no . possible triange =n(n+1)/2C3 - 3(n∑ i=3 iC3)

butx1,x4,x6,x7are also collinear...

dont know how to proceed

plz help!!!