JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Critical Thinking Questions

  • question_answer For \[2\le r\le n,\left( \begin{matrix}    n  \\    r  \\ \end{matrix} \right)+2\,\left( \begin{align}   & \,\,n \\  & r-1 \\ \end{align} \right)\]\[+\left( \begin{matrix}    n  \\    r-2  \\ \end{matrix} \right)\] is equal to    [IIT Screening 2000; Pb. CET 2000]

    A) \[\left( \begin{matrix}    n+1  \\    r-1  \\ \end{matrix} \right)\]

    B) \[2\,\left( \begin{matrix}    n+1  \\    r+1  \\ \end{matrix} \right)\]

    C) \[2\,\left( \begin{matrix}    n+2  \\    r  \\ \end{matrix} \right)\]

    D) \[\left( \begin{matrix}    n+2  \\    r  \\ \end{matrix} \right)\]

    Correct Answer: D

    Solution :

    Expression \[={{\,}^{n}}{{C}_{r}}+2\,.{{\,}^{n}}{{C}_{r-1}}{{+}^{n}}{{C}_{r-2}}\] \[={{(}^{n}}{{C}_{r}}+{{\,}^{n}}{{C}_{r-1}})+{{(}^{n}}{{C}_{r-1}}+{{\,}^{n}}{{C}_{r-2}})\] \[={{\,}^{n+1}}{{C}_{r}}+{{\,}^{n+1}}{{C}_{r-1}}={{\,}^{n+2}}{{C}_{r}}\].

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