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JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Self Evaluation Test - Permutations and Combinatioins

  • question_answer
    The sides AB, BC, CA, of a triangle ABC have 3, 4 and 5 interior points respectively on them. The total number of triangles that can be constructed by using these points as vertices is

    A) 220

    B) 204

    C) 205

    D) 195

    Correct Answer: C

    Solution :

    [c] We have in all 12 points. Since, 3 points are used to form a triangle, therefore the total number of triangles including the triangles formed by collinear points on AB, BC and CA is \[^{12}{{C}_{3}}=220.\] But this includes the following: The number of triangles formed by 3 points on \[AB=\,{{\,}^{3}}{{C}_{3}}=1\] The number of triangles formed by 4 points on \[BC{{=}^{4}}{{C}_{3}}=4.\] The number of triangles formed by 5 points on \[CA{{=}^{5}}{{C}_{3}}=10.\] Hence, required number of triangles \[=220-(10+4+1)=205.\]


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