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Math Help - Ordering people in a line

  1. #1
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    Ordering people in a line

    How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other?

    Can someone explain the solution step by step?
    Thank you. =D
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  2. #2
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    Hello, feiyingx!

    How many ways are there for eight men and five women to stand in a line
    so that no two women stand next to each other?

    This approach seems to be the easiest . . .


    Place the eight men in a row. .There are: 8! \,= \,40,320 ways.


    Leave a space between the men (and on the ends):

    . . _ M _ M _ M _ M _ M _ M _ M _ M _


    To place the five women, we have a choice of nine spaces.
    . . There are: 9\cdot8\cdot7\cdot6\cdot5 \,= \,15,\!120 choices.


    Therefore, there are: . 40,320 \times 15,120 \;= \;\boxed{609,638,400 \text{ ways.}}

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  3. #3
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    Thanks a bunch for the response. xD It was really helpful.
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