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

  • question_answer
    'n' is selected form the set \[\{1,2,3...,100\}\]and the number \[{{2}^{n}}+{{3}^{n}}+{{5}^{n}}\] is formed. Total number of ways of selecting 'n' so that the formed number is divisible by 4, is equal to

    A) 50

    B) 49

    C) 48

    D) None of these

    Correct Answer: B

    Solution :

    [b] If n is odd, \[{{3}^{n}}=4{{\lambda }_{1}}-1,{{5}^{n}}=4{{\lambda }_{2}}+1\] \[\Rightarrow {{2}^{n}}+{{3}^{n}}+{{5}^{n}}\] is not divisible by 4, as \[{{2}^{n}}+{{3}^{n}}+{{5}^{n}}\] will be in the form of \[4\lambda +2.\] Thus total number of ways of selecting ?n? is equal to 49.

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