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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 \[{{(}^{10}}{{C}_{0}})+{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}})+{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}})\] \[{{+}^{10}}{{C}_{2}})+.....+{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}}{{\,}^{10}}{{C}_{2}}+....{{+}^{10}}{{C}_{9}})\] is

    A) \[{{2}^{10}}\]

    B) \[{{10.2}^{9}}\]

    C) \[{{10.2}^{10}}\]

    D) None of these

    Correct Answer: B

    Solution :

    [b] \[{{(}^{10}}{{C}_{0}})+{{(}^{10}}{{C}_{0}}+{{\,}^{10}}{{C}_{1}})+{{(}^{10}}{{C}_{0}}+{{\,}^{10}}{{C}_{1}}+{{\,}^{10}}{{C}_{2}})\] \[+....+{{(}^{10}}{{C}_{0}}+{{\,}^{10}}{{C}_{1}}+{{\,}^{10}}{{C}_{2}}+...+{{\,}^{10}}{{C}_{9}})\] \[=10{{\,}^{10}}{{C}_{0}}+9{{\,}^{10}}{{C}_{1}}+8{{\,}^{10}}{{C}_{2}}+...+{{\,}^{10}}{{C}_{9}}\] \[={{\,}^{10}}{{C}_{1}}+2{{\,}^{10}}{{C}_{2}}+3{{\,}^{10}}{{C}_{3}}+...10{{\,}^{10}}{{C}_{10}}\] \[=\sum\limits_{r=1}^{10}{{{r}^{10}}{{C}_{r}}=10\sum\limits_{r=1}^{10}{^{9}{{C}_{r-1}}={{10.2}^{9}}}}\]


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