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How many ways are there for four men and five women ..

  1. Feb 13, 2019 at 11:08 AM #1
    1. The problem statement, all variables and given/known data
    How many ways are there for four men and five women to stand in a line so that

    All men stand together?

    All women stand together?

    2. Relevant equations


    3. The attempt at a solution
    For all men stand together, you can group the 4 men as one token, then there are (1+5)! ways the men and women can stand in a line, but the four men can be arranged in 4! ways so the answer would be (1+5)!(4!) = (6!)(4!)

    similar for women
    let 5 women = one token, then you have (1+4)! but the 5 women can be arranged 5! ways so you have (5!)(5!) as the answer

    is there anything I am missing?
     
  2. jcsd
  3. Feb 13, 2019 at 12:59 PM #2

    PeterDonis

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    Moderator's Note: Thread moved to pre-calculus math homework forum.
     
  4. Feb 13, 2019 at 7:10 PM #3

    WWGD

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    Men can go in positions 1-5, 2-6, etc.
     
  5. Feb 14, 2019 at 1:17 AM #4
    So I am taking the 4 men as one line. There are 4! possible combinations for the men. Then, There are 5 women, and the group of four men which I'm considering as 1. There are 6! possible arrangements here.

    Wouldn't "men can go in positions 1-5,2-6, etc" be under the (6!)?
     
  6. Feb 14, 2019 at 2:33 AM #5

    PeroK

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    That's right. You could have tested your approach with smaller numbers, perhaps 2 and 3.
     
  7. Feb 14, 2019 at 2:38 AM #6
    Sorry are both answers correct? WWGD's post has me a little paranoid lol
     
  8. Feb 14, 2019 at 2:49 AM #7

    PeroK

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    Yes, the answers are correct.

    That's another way to do it. For the men as a group, they have 6 possible positions (with the first man in position 1-2-3-4-5 or 6). Then it's ##6 \times 4! \times 5!##, which is the same as you got.
     
  9. Feb 14, 2019 at 5:51 PM #8

    WWGD

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    Don't worry, my reply agrees with PeroK's and yours.
     
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