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Math Help - how many ways

  1. #1
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    how many ways

    The figure shows all roads between Q(Quarryton), R(Richfield), and B(Bayview). Martina is traveling from Quarryton to Bayview and back. How many different ways could she make the round-trip, going through Richfield exactly once on a round-trip and not traveling any section of road more than once on a round trip? The ans is 12. But I don't 12~
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  2. #2
    MHF Contributor Quick's Avatar
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    I just want to say your skills at paint are somewhat terrifying. But getting on subject...

    a nice way to do this is tree diagrams...

    first draw all the combinations when going from Q to B to R back to Q
    Code:
                Q
               /
             R--- Q
            /  \ 
           /     Q
    Q-----B
           \    Q
            \  /
             R---Q
               \
                Q
    So there are 6 combinations that way

    Now draw the number of combinations from Q to R to B back to Q
    Code:
           B---Q
         /
        R---B---Q
       /    B---Q
      /    /
    Q-----R
      \    \
       \    B---Q
        R---B---Q
         \
          B---Q
    There are 6 combinations this way...

    altogether that makes 12
    Last edited by Quick; July 9th 2006 at 07:25 AM.
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  3. #3
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    Nice diagrams and great explanation, Quick!

    I sort of "eyeballed" the diagram and reasoned it out.


    Under the restrictions of the problem,
    . . there are only two routes that Martina can take.

    [1]\;\;Q\quad\longrightarrow\quad B\quad\longrightarrow\quad R \quad\longrightarrow\quad Q
    . . . . .1 way . . .2 ways . . .3 ways . . . . 1 \times 2 \times 3 \:=\:6\;ways


    [2]\;\;Q\quad\longrightarrow\quad R\quad\longrightarrow\quad B\quad\longrightarrow\quad Q
    . . . . 3 ways . . .2 ways . . .1 way . . . . 3 \times 2 \times 1 \:=\:6\;ways


    Therefore, there are: . 6 + 6 \:=\:12 ways.


    But your tree diagrams are great: . \text{1 picture} \;= \;1\text{ kiloword}

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    FYI

    1 millihelen = amount of beauty required to launch one ship.

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  4. #4
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by Soroban
    Nice diagrams and great explanation, Quick!
    Thank you, I learned how to do those code things from you! Although I can't imagine trying to make geometric shapes out of them. You know, I had such a picky 8th grade teacher that she would have pointed out that the diagrams didn't go upward
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  5. #5
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    Thanks, and I will work on the drawing
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  6. #6
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    Quote Originally Posted by Soroban
    Nice diagrams and great explanation, Quick!

    I must admit I wish I had those art coding skills, they clearly come in useful. O' Alas, the great Lord did not bless me with art
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  7. #7
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    Quote Originally Posted by Soroban
    \text{1 picture} \;= \;1\text{ kiloword}
    haha...i like that.
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