Question

The sides AB, BC, and CA of a triangle ABC have 3, 4, and 5 interior points, respectively, on them. The number of triangles that can be constructed using these interior points as vertices is ___.

Answer 1

We have total 3+4+5 = 12 points. So, number of $\Delta s$ that can be formed using 12 such points is given by
Total number of ways of selecting three point - Number of ways three collinear points are selected
${=}^{12}{C}_3{-}^3{C}_3{-}^4{C}_3{-}^5{C}_3=\frac{12\times 11\times 10}{6}-1-\frac{5\times 4}{2\times 1}=220-15=205$