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

  • question_answer
    If \[^{n}{{C}_{r}}\] denotes the number of combinations of n things taken r at a time, then the expression \[^{n}{{C}_{r+1}}+{{\,}^{n}}{{C}_{r-1}}+\,2\times {{\,}^{n}}{{C}_{r}}\] equals [AIEEE 2003]

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

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

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

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

    Correct Answer: B

    Solution :

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


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