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  4. On the sides AB , BC , CA of a triangle ABC , 3,4,5 distinct points (excluding vertices A , B , C ) are chosen respectively. The number of triangles that can be constructed using these chosen points as vertices are

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Q. On the sides $AB , BC , CA$ of a triangle $ABC , 3,4,5$ distinct points (excluding vertices $A , B , C$ ) are chosen respectively. The number of triangles that can be constructed using these chosen points as vertices are

Permutations and Combinations

Solution:

Number of triangles formed
$={ }^3 C _2 \times\left({ }^4 C _1+{ }^5 C _1\right)+{ }^4 C _2 \times\left({ }^3 C _1+{ }^5 C _1\right)+{ }^5 C _2 \times\left({ }^3 C _1+{ }^4 C _1\right)+{ }^3 C _1 \times{ }^4 C _1 \times{ }^5 C _1 $
$=27+48+70+60=205$