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Math Help - permutation question

  1. #1
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    permutation question

    Using the letters GOOGOOGAGAGA determine the number of distinguishable arrangements with the first O after the first G but not necessarily right after?

    G=5
    0=4
    A=3

    I know the total possibilities are 12!/(5!4!3!) but that would include an O coming before a G.
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  2. #2
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    Hello, jloco1991!

    I think I have approach to this problem.
    If I'm wrong, someone will certainly say so.


    Using the letters GOOGOOGAGAGA determine the number of distinguishable
    arrangements with the first O after the first G, but not necessarily right after?
    . . (Five G's, four O's, three A's)

    I see that there are four cases to consider.


    (1) The first \,G is the first letter: . G\,\+\,\_\,\_\,\_\,\hdots

    . . .The other 11 letters \{G,G,G,G,O,O,O,O,A,A,A\}

    . . . . .can be arranged in: . \displaystyle{11\choose4,4,3} \:=\:11,\!550\text{ ways.}


    (2) The first \,G is the second letter: . A\,G\,\_\,\_\,\_\,\hdots

    . . .The other 10 letters \{G,G,G,G,O,O,O,O,A,A\}

    . . . . .can be arranged in \displaystyle{10\choose4,4,2} \:=\: 3,\!150\text{ ways.}


    (3) The first \,G is the third letter: . A\,A\,G\,\_\,\_\,\_\,\hdots

    . . .The other 9 letters \{G,G,G,G,O,O,O,O,A\}

    . . . . . can be arranged in \displaystyle{9\choose4.4.1} \:=\:630\text{ ways.}


    (4) The first \,G is the fourth letter: . A\,A\,A\,G\,\_\,\_\,\_\,\hdots

    . . .The other 8 letters \{G,G,G,G,O,O,O,O\}

    . . . . . can be arranged in \displaystyle{8\choose4,4} \:=\:70\text{ ways.}


    Therefore, there are:

    . . 11,\!550 + 3,\!150 + 630 + 70 \:=\:15,\!400\text{ arrangements.}

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