Number of triangles

[hey_thr67]

The side of BC,CA, and AB of a triangle have 3,4, and 5 points respectively on them. The no. of triangles that can be constructed using these points as vertices is

a) 200
b) 205
c) 400
d) 410
e) 820

[mathbyvemuri]

total number of points = 3+4+5 = 12
If all the 12 points are not collinear we can have 12C3 triangles. (a triangle to be formed requires three non-collinear points)
But we have three sets of collinear points (3, 4 and 5 points in the sets)
So we need to subtract the number of triangles formed by these sets.
Tot number of triangles = 12C3 - 3C3 - 4C3 - 5C3 = 205
Answer b