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Math Help - Re-arranging letters in a word...

  1. #1
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    Re-arranging letters in a word...

    This is my solution to the following question. Case 3, seems to be wrong though.

    How many 4 letter arrangements of the word TOMORROW are there?

    So i separate letters into their own "buckets": T, OOO, M, RR, W

    So here are the cases:

    Case1: 4 letters are all different : 5C4 * 4!
    Case2: 2 Letters are same, 2 Letters are different: 2C1 * 4C2 * 4!/2!
    Case3: 2 letters are same, 2 letters are same: 2C1 * 1C1 * 4!/2!
    Case4: 3 Same, 1 Different: 1C1 * 4 C 1 * 4!/3!

    Add them all up at the end and you get the answer.

    But did i do Case3 right?
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, Saibot!

    This is my solution to the following question. Case 3, seems to be wrong though.

    How many 4-letter arrangements of the word TOMORROW are there?

    So i separate letters into their own "buckets": T, OOO, M, RR, W

    So here are the cases:

    Case 1: 4 letters are all different: . \left(_5C_4\right)(4!)

    Case 2: 2 Letters are same, 2 Letters are different: . \left(_2C_1\right)\left(_4C_2\right)\left(\frac{4!  }{2!}\right)

    Case 3: 2 letters are same, 2 letters are same: . \left(_2C_1\right)\left(_1C_1\right)\left(\frac{4!  }{2!}\right)

    Case 4: 3 Same, 1 Different: . \left(_1C_1\right)\left(_4C_1\right)\left(\frac{4!  }{3!}\right)

    Add them all up at the end and you get the answer.

    But did i do Case 3 right?

    Case 3: Two letters are same, two letters are same.

    The four letters must be: O,O,R,R.

    And there are: . \frac{4!}{2!2!} arrangements.

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