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JEE Main & Advanced Sample Paper JEE Main Sample Paper-10

  • question_answer
    If \[n\in N\]and\[{{C}_{r}},{{C}_{r-1,}}{{C}_{r-2}}\]usual meaning, then the value of the expression\[^{n-2}{{C}_{r}}+{{2}^{n-2}}{{C}_{r-1}}{{+}^{n-2}}{{C}_{r-2}}\] equals

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

    B)  \[^{n}{{C}_{r-1}}\]

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

    D)  \[^{n}{{C}_{r}}\]

    Correct Answer: D

    Solution :

     The given expression is \[^{n-2}{{C}_{r}}+{{2}^{n-2}}{{C}_{r-1}}{{+}^{n-2}}{{C}_{r-2}}\] \[={{(}^{n-2}}{{C}_{r}}{{+}^{n-2}}{{C}_{r-1}})+{{(}^{n-2}}{{C}_{r-1}}{{+}^{n-2}}{{C}_{r-2}})\] \[{{=}^{n-2+1}}{{C}_{r}}{{+}^{n-2+1}}{{C}_{r-1}}\] \[{{=}^{n-1}}{{C}_{r}}{{+}^{n-1}}{{C}_{r-1}}\] \[{{=}^{(n-1+1)}}{{C}_{r}}{{=}^{n}}{{C}_{r}}\]So, correct option is [d].


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