Question

$18$ points are indicated on the perimeter of a triangle $ABC$ (see figure). How many triangles are there with vertices at these points?

• A. $331$
• B. $408$
• C. $710$
• D. $711$

Answer 1

The correct option is D $711$

18 points on the perimeter of triangle ABC.

$\to$ Total number of triangle ${=}^{18}{C}_3$

$\to$ but the triangle can not be formed by joining the

the point which uses in a straight line.

So possible triangle ${=}^{18}{C}_3-3{\times }^7{C}_3$

$=\frac{18\times 17\times 16}{3\times 2\times 1}-3\times \frac{7\times 6\times 5}{3\times 2\times 1}$

$=816-105$

$=711$