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JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Self Evaluation Test - Binomial Theorem

  • question_answer
    The value of \[^{20}{{C}_{0}}+{{\,}^{20}}{{C}_{1}}+{{\,}^{20}}{{C}_{2}}+{{\,}^{20}}{{C}_{3}}+{{\,}^{20}}{{C}_{4}}\]\[+{{\,}^{20}}{{C}_{12}}+{{\,}^{20}}{{C}_{13}}+{{\,}^{20}}{{C}_{14}}+{{\,}^{20}}{{C}_{15}}\] is

    A) \[{{2}^{19}}-\frac{\left( ^{20}{{C}_{10}}+{{\,}^{20}}{{C}_{9}} \right)}{2}\]

    B) \[{{2}^{19}}-\frac{\left( ^{20}{{C}_{10}}+\,2{{\times }^{20}}{{C}_{9}} \right)}{2}\]

    C) \[{{2}^{19}}-\frac{^{20}{{C}_{10}}}{2}\]

    D) None of these

    Correct Answer: B

    Solution :

    [b] Given series is \[^{20}{{C}_{0}}+{{\,}^{20}}{{C}_{1}}+{{\,}^{20}}{{C}_{2}}+...+{{\,}^{28}}{{C}_{8}}\] \[=\frac{({{2}^{20}}-{{\,}^{20}}{{C}_{10}})}{2}-{{\,}^{20}}{{C}_{9}}\] \[={{2}^{19}}-\frac{({{\,}^{20}}{{C}_{10}}+2\times {{\,}^{20}}{{C}_{9}})}{2}\]


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