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JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

  • question_answer
    The coefficient of \[{{x}^{53}}\] in the following expansion  \[\sum\limits_{m=0}^{100}{{{\,}^{100}}{{C}_{m}}{{(x-3)}^{100-m}}}{{.2}^{m}}\]is

    A) \[^{100}{{C}_{47}}\]

    B) \[^{100}{{C}_{53}}\]

    C) \[{{-}^{100}}{{C}_{53}}\]

    D) \[{{-}^{100}}{{C}_{100}}\]

    Correct Answer: C

    Solution :

    The given sigma is expansion of \[{{[(x-3)+2]}^{100}}={{(x-1)}^{100}}={{(1-x)}^{100}}\]. \ \[{{x}^{53}}\]will occur in \[{{T}_{54}}\].    \[{{T}_{54}}={{\,}^{100}}{{C}_{53}}{{(-x)}^{53}}\] \ Coefficient is - 100C53.


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