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JEE Main & Advanced Sample Paper JEE Main Sample Paper-20

  • question_answer
    The expansion of \[{{(1+x)}^{n}}\] has \[3\] consecutive terms with coefficients in the ratio \[1:2:3\] and can be written in the form\[^{n}{{C}_{k}}{{:}^{n}}{{C}_{k+1}}{{:}^{n}}{{C}_{k+2}}\].The sum of all Possible values of\[(n+k)\] is-

    A) \[18\]                                   

    B) \[21\]

    C) \[28\]                                   

    D) \[32\]

    Correct Answer: A

    Solution :

    \[\frac{^{n}{{C}_{k}}}{^{n}{{C}_{k+1}}}=\frac{1}{2}\Rightarrow \frac{n!}{k!(n-k)!}=\frac{(k+1)!(n-k-1)!}{n!}=\frac{1}{2}\] or \[\frac{k+1}{n-k}=\frac{1}{2}\]                 \[2k+2=n-k\]                 \[n-3k=2\]                                  .......... (1) Similarly,              \[\frac{^{n}{{C}_{k+1}}}{^{n}{{C}_{k+2}}}=\frac{2}{3}\] \[\frac{n!}{(k+1)!(n-k-1)!}\cdot \frac{(k+2)!(n-k-2)!}{n!}=\frac{2}{3}\]                 \[\frac{k+2}{n-k-1}=\frac{2}{3}\]                 \[2n-5k=8\]                             ............. (2) From (1) and (2)                        \[n=14\]and\[k=4\] \[\therefore \]  \[n+k=18\]


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