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JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Self Evaluation Test - Permutations and Combinatioins

  • question_answer
    The expression \[^{n}{{C}_{r}}+{{4.}^{n}}{{C}_{r-1}}+{{6.}^{n}}{{C}_{r-2}}+{{4.}^{n}}{{C}_{r-3}}\]\[{{+}^{n}}{{C}_{r-4}}\] is equal to

    A) \[^{n+4}{{C}_{r}}\]

    B) \[{{2.}^{n+4}}{{C}_{r-1}}\]

    C) \[{{4.}^{n}}{{C}_{r}}\]

    D) \[{{11.}^{n}}{{C}_{r}}\]

    Correct Answer: A

    Solution :

    [a] \[^{n}{{C}_{7}}+{{4.}^{n}}{{C}_{r-1}}+{{6.}^{n}}{{C}_{r-2}}+{{4.}^{n}}{{C}_{r-3}}{{+}^{n}}{{C}_{r-4}}\]
    \[={{(}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r-1}})+3{{(}^{n}}{{C}_{r-1}}{{+}^{n}}{{C}_{r-2}})\]
    \[+3{{(}^{n}}{{C}_{r-2}}{{+}^{n}}{{C}_{r-3}})+{{(}^{n}}{{C}_{r-3}}{{+}^{n}}{{C}_{r-4}})\]
    \[=n{{+}^{1}}{{C}_{r}}+{{3.}^{n+1}}{{C}_{r-1}}+{{3.}^{n+1}}{{C}_{r-2}}{{+}^{n+1}}{{C}_{r-3}}\]
    \[={{(}^{n+1}}{{C}_{r}}{{+}^{n+1}}{{C}_{r-1}})+2{{(}^{n+1}}{{C}_{r-1}}+\]
    \[+{{(}^{n+1}}{{C}_{r-2}}{{+}^{n+1}}{{C}_{r-3}})\]
    \[{{=}^{n+2}}{{C}_{r}}+2.{{(}^{n+2}}{{C}_{r-1}}{{+}^{n+2}}{{C}_{r-2}})\]
    \[={{(}^{n+2}}{{C}_{r}}{{+}^{n+2}}{{C}_{r-1}})+{{(}^{n+2}}{{C}_{r-1}}{{+}^{n+2}}{{C}_{r-2}})\]
    \[{{=}^{n+3}}{{C}_{r}}{{+}^{n+3}}{{C}_{r-1}}{{=}^{n+4}}{{C}_{r}}\]


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