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Combinatorics, permutations of letters

  1. Dec 17, 2012 #1
    1. The problem statement, all variables and given/known data
    How many "words" with 5 letters can be created from the letters in the word ALGEBRA? Each letter can be used only once.


    2. Relevant equations



    3. The attempt at a solution
    I know the answer to this (1320) and I know how I got to it (which I'll describe in a minute), but I know that while my way of getting it is right, it's. not the easiest, and I can't seem to figure out how on earth they want me to do it.

    I know that choosing 5 things from a set with 7 elements can be done in

    [itex]\frac{7!}{(7-5)!}[/itex]

    ways. I also know that if I were to use all the letters of ALGEBRA the answer would be

    [itex]\frac{7!}{2!}[/itex]

    since one letter is used twice. In this case these happen to be exactly same, but I can't seem to put these principles together to get a correct answer. Help, please?

    The way I solved it was more of a brute force solution. The correct answer is 11 * 5!, and I got to that conclusion by reasoning that 5! of the words are made by the letters LGEBR, which have no recurring letters. Then I "exchanged" one of the letters from that to an A, one at a time, and then to both As. This can be done in 11 ways:

    LGEBR
    AGEBR
    LAEBR
    LGABR
    LGEAR
    LGEBA
    AAEBR
    LAABR
    LGAAR
    LGEAA
    AGEBA

    And all of them can be chosen in 5! different combinations. I know this is the correct answer (it's an online based homework, I've inputted this and it says I'm right).
     
  2. jcsd
  3. Dec 17, 2012 #2

    tiny-tim

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    Hi Hannisch! :smile:

    That certainly works.

    But quicker would be to split the problem into three …

    count separately the number of words with no As, with one A, and with 2As. :wink:
     
  4. Dec 17, 2012 #3

    haruspex

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    Slightly quicker still, just separate the cases "at most one A" (just as with all the other letters) and "2 As"
     
  5. Dec 18, 2012 #4

    tiny-tim

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    oh yes! :biggrin:
     
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