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JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Definition of combinations, Conditional combinations, Division into groups, Derangements

  • question_answer
    \[^{n}{{C}_{r}}+{{2}^{n}}{{C}_{r-1}}{{+}^{n}}{{C}_{r-2}}=\]

    A) \[^{n+1}{{C}_{r}}\]

    B) \[^{n+1}{{C}_{r+1}}\]

    C) \[^{n+2}{{C}_{r}}\]

    D) \[^{n+2}{{C}_{r+1}}\]

    Correct Answer: C

    Solution :

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


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