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JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Geometrical Problems

  • question_answer
    Let \[{{T}_{n}}\] denote the number of triangles which can be formed using the vertices of a regular polygon of \[n\] sides. If \[{{T}_{n+1}}-{{T}_{n}}=21,\] then \[n\] equals [IIT Screening 2001]

    A) 5

    B) 7

    C) 6

    D) 4

    Correct Answer: B

    Solution :

    Clearly, \[^{n}{{C}_{3}}\,={{T}_{n}}.\]  So, \[^{n+1}{{C}_{3}}\,-{{\,}^{n}}{{C}_{3}}\,=21\] Þ \[(\,{{\,}^{n}}{{C}_{3}}\,+{{\,}^{n}}{{C}_{2}}\,)\,-{{\,}^{n}}{{C}_{3}}\,=21\] \ \[^{n}{{C}_{2}}\,=\,21\] or\[n(n-1)=42=7\,.\,6\,\,\,\,\therefore \,\,\,n=7\].


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