JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Critical Thinking Questions

  • question_answer \[m\] men and n women are to be seated in a row so that no two women sit together. If \[m>n\], then the number of ways in which they can be seated is [IIT 1983]

    A) \[\frac{m\ !\ (m+1)\ !}{(m-n+1)\ !}\]

    B) \[\frac{m\ !\ (m-1)\ !}{(m-n+1)\ !}\]

    C) \[\frac{(m-1)\ !\ (m+1)\ !}{(m-n+1)\ !}\]

    D) None of these

    Correct Answer: A

    Solution :

    First arrange m men, in a row in m! ways. Since \[n<m\] and no two women can sit together, in any one of the \[m\,!\] arrangement, there are \[(m+1)\] places in which n women can be arranged in \[^{m+1}{{P}_{n}}\] ways. \[\therefore \,\]By the fundamental theorem, the required number of arrangements of m men and n women \[(n<m)\] = \[m\,!{{.}^{m+1}}{{P}_{n}}=\frac{m\,!.(m+1)!}{\{(m+1)-n\}\,!}\,=\frac{m!(m+1)!}{(m-n+1)\,!}\].

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