Choose 3, 4, 5 points other than vertices respectively on the sides AB, BC and CA of a triangle ABC. The number of triangles that can be formed using only these points as vertices is

Question

Choose 3, 4, 5 points other than vertices respectively on the sides AB, BC and CA of a triangle ABC. The number of triangles that can be formed using only these points as vertices is (A) 220 (B) 217 (C) 215 (D) 210

Tardigrade Admin · Accepted Answer

Required number of triangles that can be formed by using only given points as vertices = 12 C3- 3 C3+ 4 C3+ 5 C3 =(12 × 11 × 10/3 × 2 × 1)- 1+4+(5 × 4/2 × 1) =220-(5+10) =220-15=205

Q. Choose $3, 4, 5$ points other than vertices respectively on the sides $A B, BC$ and $C A$ of a triangle $A BC$ . The number of triangles that can be formed using only these points as vertices is

220

217

215

210

205

Required number of triangles that can be formed by using only given points as vertices
$=^{12} C_{3} - {^{3} C_{3} +^{4} C_{3} +^{5} C_{3}}$
$= \frac{12 \times 11 \times 10}{3 \times 2 \times 1} - {1 + 4 + \frac{5 \times 4}{2 \times 1}}$
$= 220 - (5 + 10)$
$= 220 - 15 = 205$

Q. Choose 3,4,5 points other than vertices respectively on the sides AB,BC and CA of a triangle ABC. The number of triangles that can be formed using only these points as vertices is

220

217

215

210

205

Required number of triangles that can be formed by using only given points as vertices =12C3​−{3C3​+4C3​+5C3​} =3×2×112×11×10​−{1+4+2×15×4​} =220−(5+10) =220−15=205

Q. Choose $3, 4, 5$ points other than vertices respectively on the sides $A B, BC$ and $C A$ of a triangle $A BC$ . The number of triangles that can be formed using only these points as vertices is

Required number of triangles that can be formed by using only given points as vertices
$=^{12} C_{3} - {^{3} C_{3} +^{4} C_{3} +^{5} C_{3}}$
$= \frac{12 \times 11 \times 10}{3 \times 2 \times 1} - {1 + 4 + \frac{5 \times 4}{2 \times 1}}$
$= 220 - (5 + 10)$
$= 220 - 15 = 205$