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

  • question_answer
    If \[S=(1)(1!)+(2)(2!)+(3)(3!)+...+n(n!),\] then

    A) \[\frac{S+1}{n!}\in \]integer

    B) \[\frac{S+1}{n!}\notin \] Integer

    C) \[\frac{S+1}{n!}\] cannot be discussed

    D) None of these

    Correct Answer: A

    Solution :

    [a] We have, \[S=\sum\limits_{k=1}^{n}{k(k!)}=\sum\limits_{k=1}^{n}{\{(k+1)-1\}}(k!)\] \[=\sum\limits_{k=1}^{n}{\{(k+1)!-k!\}=(n+1)!-1\Rightarrow S+1=(n+1)!}\] Thus, \[\frac{S+1}{n!}\in \] integer.

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