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

  • question_answer
    The value of ?n? for which \[^{n-1}{{C}_{4}}{{-}^{n-1}}{{C}_{3}}-\frac{5}{4}{{.}^{n-1}}{{P}_{2}}<0,\] Where \[n\in N\]

    A) \[\{5,6,7,8,9,10\}\]

    B) \[\{1,2,3,4,5,6,7,8,9,10\}\]

    C) \[\{1,4,5,6,7,8,9,10\}\]

    D) \[(-\infty ,2)\cup (3,11)\]

    Correct Answer: A

    Solution :

    [a] we have, \[^{n-1}{{C}_{4}}{{-}^{n-1}}{{C}_{3}}-\frac{5}{4}{{,}^{n-2}}{{P}_{2}}<0\]
    \[\Rightarrow \frac{(n-1)(n-2)(n-3)(n-4)}{4!}-\frac{(n-1)(n-2)(n-3)}{3!}\]
    \[\Rightarrow (n-2)(n-3)(n-11)(n+2)<0\]
    \[\Rightarrow (n-2)(n-3)(n-11)<0\]
    \[[\because n+2>0forn\in N]\]
    \[\Rightarrow n\in (-\infty ,2)\cup (3,11)\]
    \[\Rightarrow n\in (0,2)\cup (3,11)\]
    \[\Rightarrow n=1,4,5,6,7,8,9,10\]
    But \[^{n-1}{{C}_{4}}\] and \[^{n-2}{{P}_{2}}\] both are meaningful for \[n\ge 5.\]

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