If the sides A B , B C and C A of a triangle A B C have 3 , 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

Step-by-step solution

We know that for making a triangle, we require three points, which are not on a line. Here, we have total 3 + 5 + 6 = 14 points out of which 3 ,   5 ,   6 points respectively, on A B ,   B C and C A are collinear. Thus, the total Number of triangles formed = C 3 14 - C 3 3 - C 3 5 - C 3 6 = 14 ! 3 ! · 11 ! - 3 ! 3 ! · 0 ! - 5 ! 3 ! · 2 ! - 6 ! 3 ! · 3 ! = 14 × 13 × 12 × 11 ! 3 × 2 × 1 × 11 ! - 1 - 5 × 4 × 3 ! 3 ! × 2 × 1 - 6 ×