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Find the number of possible triangles using points on the sides of any triangle ABC having “a” points on side BC, “b” points on side AC, “c” points on side AB excluding the points at vertices.

Find the number of possible triangles using points on the sides of any triangle ABC having “a” points on side BC, “b” points on side AC, “c” points on side AB excluding the points at vertices. Correct Answer <sup>a + b + c</sup>C<sub>3</sub> – <sup>a</sup>C<sub>3</sub> – <sup>b</sup>C<sub>3</sub> – <sup>c</sup>C<sub>3</sub>

Number of triangles formed using points a, b and c = Selecting 3 points out of a, b and c

⇒ Number of triangles formed using points a, b and c = a + b + cC3

But this also includes those 3 points which lie on the side BC and if we take these 3 points we get a straight line instead of any triangle

So applying similar concept we subtract such cases on lines AC and AB also

⇒ Number of possible triangles = a + b + cC3 – aC3 – bC3 – cC3

∴ Number of possible triangles = a + b + cC3 – aC3 – bC3 – cC3

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