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

  • question_answer
    If \[x+y=1\], then \[\sum\limits_{r=0}^{n}{{{r}^{n}}{{C}_{r}}{{x}^{r}}{{y}^{n-r}}}\] equals

    A) 1

    B) n

    C) nx

    D) ny

    Correct Answer: C

    Solution :

    [c] We have, \[\sum\limits_{r=0}^{n}{r{{\,}^{n}}{{C}_{r}}{{x}^{r}}\,{{y}^{n-r}}=\sum\limits_{r=0}^{n}{r\frac{n}{r}{{\,}^{n-1}}{{C}_{r-1}}{{x}^{r-1}}{{x}^{1}}{{y}^{n-r}}}}\] \[=\,\,nx\,\,\sum\limits_{r=0}^{n}{^{n-1}{{C}_{r-1}}{{x}^{r-1}}{{y}^{(n-1)-(r-1)}}}\] \[=\,\,\,\,nx{{(x+y)}^{n-1}}=nx\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,[\because \,\,\,\,x+y=1]\]


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