There are 5 points \(P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\) on the side AB, excluding A and B, of a triangle ABC. Similarly there are 6 points \(P_{6}, P_{7}, \ldots, P_{11}\) on the side BC and 7 points \(P_{12}, P_{13}, \ldots, P_{18}\) on the side CA of the triangle. The number of triangles, that can be formed using the points \(P_{1}, P_{2}, \ldots, P_{18}\) as vertices, is

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Permutations and Combinations

There are 5 points

P_{1}, P_{2}, P_{3}, P_{4}, P_{5}

on the side AB, excluding A and B, of a triangle ABC. Similarly there are 6 points

P_{6}, P_{7}, \dots, P_{11}

on the side BC and 7 points

P_{12}, P_{13}, \dots, P_{18}

on the side CA of the triangle. The number of triangles, that can be formed using the points

P_{1}, P_{2}, \dots, P_{18}

as vertices, is

Answer: (D)

Total ways to choose 3 of the 18 points:

(3 18) = 816

. Subtract collinear triples on each side (

5

A B

6

B C

7

C A

816 - (3 5) - (3 6) - (3 7) = 816 - 10 - 20 - 35 = 751.

Option D.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Single correct option