NDA 1 2024 Maths, question 6

Topic: Algebra

Q6

A triangle $PQR$ is such that $3$ points lie on the side $PQ$ , $4$ points on $QR$ and $5$ points on $RP$ respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed ?

(a) $205$ Correct answer
(b) $206$
(c) $215$
(d) $220$

Solution.

Answer: (a) $205$

Choose three of the 12 marked points, then remove triples lying on one side: C(12,3)−C(3,3)−C(4,3)−C(5,3)=220−1−4−10=205.
This uses the usual reading that the marked points are distinct interior points of the sides; a line other than a side cannot meet three distinct boundary points of a triangle.