JEE Main & Advanced Sample Paper JEE Main Sample Paper-19

  • question_answer 65) The value of\[\sum\limits_{r=1}^{5}{\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r-1}}}=}\]

    A) \[5(n-3)\]                           

    B) \[5(n-2)\]

    C) \[5n\]                                   

    D) \[5(2n-9)\]

    Correct Answer: B

    Solution :

    \[\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r-1}}}=\frac{r.\left| \!{\nderline {\,   n \,}} \right. }{\left| \!{\nderline {\,   r \,}} \right. .\left| \!{\nderline {\,   n-r \,}} \right. }\cdot \frac{\left| \!{\nderline {\,   r-1 \,}} \right. \left| \!{\nderline {\,   n-r \,}} \right. +1}{\left| \!{\nderline {\,   n \,}} \right. }=\frac{\left| \!{\nderline {\,   n-r \,}} \right. +1}{\left| \!{\nderline {\,   n-r \,}} \right. }\] \[=\frac{\left| \!{\nderline {\,   n-r \,}} \right. +1}{\left| \!{\nderline {\,   n-r \,}} \right. }=n-r+1\] \[\therefore \]\[\sum\limits_{r=1}^{5}{=}n+(n-1)+(n-2)+(n-3)+(n-4)\]                 \[=5n-10=5(n-2)\]

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