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  4. 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

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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

KEAMKEAM 2013Permutations and Combinations

Solution:

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