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Math Help - permutations problem

  1. #1
    Junior Member Bartimaeus's Avatar
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    permutations problem

    from the word STUDIO, how many arrangements are possible if the letter O is not the first or the last?

    please show working, thanks!!!
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  2. #2
    Eater of Worlds
    galactus's Avatar
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    Put the O in the first place and arrange the other 5

    Put the O in the last place and arrange the other 5

    Add them and subtract from the total number of arrangements.
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  3. #3
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    Hello, Bartimaeus!

    Another approach . . .


    From the word STUDIO, how many arrangements are possible
    if the letter O is not the first or the last?

    If the O is not first or last, there are 4 places for it: .  X\;\_ \;\_ \;\_ \;\_ \;X

    Then the other 5 letters can be placed in: . {\bf5!} ways.

    Therefore, there are: . 4\cdot5! \:=\:480 arrangements.

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  4. #4
    Eater of Worlds
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    I got the same thing by doing it the way I mentioned.

    Place the O on the end and arrange the other 5 in 120 ways.

    Place the O in the front and arrange the other 5 in 120 ways.

    That is 240 ways.

    There are 6!=720 arrangements altogether.

    So, 720-240=480 ways
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