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

  • question_answer
    The coefficient of \[{{x}^{53}}\] in the 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 :

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


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