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JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Definition of permutation, Number of permutations with or without repetition, Conditional permutations

  • question_answer
    If \[a\] denotes the number of permutations of \[x+2\] things taken all at a time, \[b\] the number of permutations of \[x\] things taken 11 at a time and \[c\] the number of permutations of \[x-11\] things taken all at a time such that \[a=182\ bc\], then the value of \[x\] is

    A) 15

    B) 12

    C) 10

    D) 18

    Correct Answer: B

    Solution :

    We have \[a{{=}^{x+2}}{{P}_{x+2}}=(x+2)\ !,\ b{{=}^{x}}{{P}_{11}}=\frac{x\ !}{(x-11)\ !}\] and \[c{{=}^{x-11}}{{P}_{x-11}}=(x-11)\ !\] Now \[a=182\ bc\Rightarrow (x+2)\ !=182\ .\ \frac{x\ !}{(x-11)\ !}(x-11)\ !\] \[\Rightarrow \ (x+2)\,!\ =182x\ !\Rightarrow (x+2)(x+1)=182\]\[\Rightarrow \]\[x=12\].


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