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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2005

  • 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:

    A)  5                                            

    B)  7

    C)  6                                            

    D)  4

    Correct Answer: B

    Solution :

                    \[{{T}_{n}}{{=}^{n}}{{C}_{3}}\Rightarrow {{T}_{n+1}}-{{T}_{n}}=21\] \[\Rightarrow \]               \[^{n+1}{{C}_{3}}{{-}^{n}}{{C}_{3}}=21\] \[\Rightarrow \]               \[^{n}{{C}_{2}}{{+}^{n}}{{C}_{3}}{{-}^{n}}{{C}_{3}}=21\] \[\Rightarrow \]               \[^{n}{{C}_{2}}=21\] \[\Rightarrow \]               \[\frac{n(n-1)}{2}=21\] \[\Rightarrow \]               \[{{n}^{2}}-n-42=0\] \[\Rightarrow \]               \[(n-7)(n+6)=0\] \[\therefore \]  \[n=7\] As           \[n\ge 1\]


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