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Math Help - Permutations of Bottles

  1. #1
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    Permutations of Bottles

    In how many different ways can I arrange 7 green and 8 brown bottles so that exactly one pair of green bottles are side by side?

    I've been trying to do this all day, am I right for separating the answers into 14 cases of where the 2 green bottles can be across 15 places?
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  2. #2
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    Hello, mixtapevanity!

    In how many different ways can I arrange 7 green and 8 brown bottles
    so that exactly one pair of green bottles are side by side?
    Duct-tape two green bottles together.

    Now we have 14 units to arrange:
    . . \boxed{GG}\,, G, G, G, G, G, B, B, B, B, B, B,B,B

    Now place the eight brown bottles in a row.
    . . Note that there are spaces before, after and between them.

    . . . \_\,B\,\_\,B\,\_\,B\,\_\,B\,\_\,B\,\_\,B\,\_\,B\,\  _\,B\,\_


    We will take the six green units and place them in six of the nine spaces.

    And there are: . {8\choose6} \:=\:\boxed{84} ways.

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  3. #3
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    Thanks for your reply! That is along the lines of what I was thinking. But can I ask what did you mean by (8 6) = 84?
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  4. #4
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    That is a mere typo. It should be \binom{9}{6}=84
    Nine blanks choosing six can be done in 84 ways.
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