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