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Thread: Sock Probability

  1. #1
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    Sock Probability

    3 pairs of socks are placed side-by-side of a straight clothes line. The socks in each pair are identical, but the pairs themselves are of different color. If I stand on one side of the line, how many different color arrangements can I see if no sock is next to its mate?
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  2. #2
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    Hello, sweeetcaroline!

    I found no elegant approach to this problem.
    I was forced to use "brute force" listing . . . *blush*


    3 pairs of socks are placed side-by-side of a straight clothes line.
    The socks in each pair are identical, but the pairs themselves are of different color.
    If I stand on one side of the line, how many different color arrangements can I see
    if no sock is next to its mate?
    Suppose the colors are Red, Blue, and Green.

    Then we have: .$\displaystyle \{R,R,B,B,G,G\}$


    We have 3 choices for the color of the first sock (#1).
    Suppose it is Red.
    The other Red can be in positions #3, 4, 5 or 6.

    Reds in (1,3): .$\displaystyle R\:\_\:R\:\_\:\_\:\_ $
    Then there are two cases:
    . . #2 is $\displaystyle B$, and the last three must be $\displaystyle GBG.$
    . . #2 is $\displaystyle G$, and the last three must be $\displaystyle BGB.$

    Reds in (1,4): .$\displaystyle R\:\_\:\_\:R\:\_\:\_ $
    Then there are four cases:
    . . #2, 3 are $\displaystyle BG$, and the last two are $\displaystyle BG$ or $\displaystyle GB.$
    . . #2, 3 are $\displaystyle GB$, and the last two are $\displaystyle BG$ or $\displaystyle GB.$

    Reds in (1,5): .$\displaystyle R\:\_\:\_\:\_\:R\:\_$
    Then there are two cases:
    . . #2, 3, 4 are $\displaystyle BGB$, and the last is $\displaystyle G.$
    . . #2, 3, 4 are $\displaystyle GBG$, and the last is $\displaystyle B.$

    Reds in (1,6): .$\displaystyle R\:\_\:\_\:\_\:\_\:R$
    Then there are two cases:
    . . #2, 3, 4, 5 are $\displaystyle BGBG.$
    . . #2, 3, 4, 5 are $\displaystyle GBGB.$

    Hence, there are: .$\displaystyle 2 + 4 + 2 + 2 \:=\:10$ arrangements that begin with Red.


    Since there are 3 choices for the color of sock #1,
    . . there are: .$\displaystyle 3 \times 10 \:=\:\boxed{30}$ arrangements with no adjacent mates.

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