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JEE Main & Advanced Sample Paper JEE Main Sample Paper-6

  • question_answer
    If \[^{n}{{C}_{0}}{{,}^{n}}{{C}_{1}}{{,}^{n}}{{C}_{2}},\]????, \[^{n}{{C}_{n}}\]denote the binomial coefficients in the expansion of \[{{(1+x)}^{n}}\] and p + q = 1, then \[\sum\limits_{r=0}^{n}{{{r}^{n}}{{C}_{r}}{{p}^{r}}{{q}^{n-r}}}\]is

    A)  np2                                       

    B)  npq

    C)  np                                         

    D)  None of these

    Correct Answer: C

    Solution :

    We have, \[\sum\limits_{r=0}^{n}{{{r}^{n}}{{C}_{r}}{{p}^{r}}{{q}^{n-r}}}\] \[=\sum\limits_{r=0}^{n}{r}.{{\frac{n}{r}}^{n-1}}{{C}_{r-1}}p.{{p}^{r-1}}{{q}^{(n-1)-(r-1)}}\] \[=np\left\{ \sum\limits_{r=0}^{n}{^{n-1}}{{C}_{r-1}}{{p}^{r-1}}{{q}^{(n-1)-(r-1)}} \right\}\] \[=np{{(q+p)}^{n-1}}\left[ \because {{(q+q)}^{n}}=\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{p}^{r}}{{q}^{n-r}}} \right]\] \[=np\]                 \[(\because p+q+1)\]


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