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Math Help - Arranging sequences of letters in a word

  1. #1
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    Arranging sequences of letters in a word

    I'm not sure if the answer I got for this problem is correct:

    a) How many distinguishable ways can the letters in ABSTEMIOUS be rearranged? (The current arrangement counts as one way.)
    b) How many of the rearrangements you found leave the vowels in alphabetical order?

    For (a), there is a total of 10 letters, but there are 2 S's, so the answer should be 10!/2! = 1814400.
    For (b), I don't want to lose the order of the vowels, so the answer would be 10!/(5!2!) = 15120. Is this reasoning correct?
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  2. #2
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    Yes those are both correct.
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    More trivia . . .


    ABSTEMIOUS is a word with the five vowels in alphabetical order.

    Another is FACETIOUS.

    Do you know the others?

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  4. #4
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    Quote Originally Posted by uberbandgeek6 View Post
    I'm not sure if the answer I got for this problem is correct:

    a) How many distinguishable ways can the letters in ABSTEMIOUS be rearranged? (The current arrangement counts as one way.)
    b) How many of the rearrangements you found leave the vowels in alphabetical order?

    For (a), there is a total of 10 letters, but there are 2 S's, so the answer should be 10!/2! = 1814400.
    For (b), I don't want to lose the order of the vowels, so the answer would be 10!/(5!2!) = 15120. Is this reasoning correct?
    Yes, keeping the vowels "in order" works just "as if" the letters are identical.
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  5. #5
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    Ha! Download the list of English words, then
    Code:
    $ grep '^[^aeiou]*a[^aeiou]*e[^aeiou]*i[^aeiou]*o[^aeiou]*u[^aeiou]*$' words.txt
    abstemious
    abstemiously
    abstentious
    acheilous
    acheirous
    acleistous
    affectious
    annelidous
    arsenious
    arterious
    bacterious
    caesious
    facetious
    facetiously
    fracedinous
    majestious
    Not all of these words are in Merriam-Webster Dictionary, though. Some are in Oxford English Dictionary or in Wiktionary. In fact, Wikipedia says this.
    There are many words that feature all five regular vowels occurring only once in alphabetical order, the most common being abstemious and facetious. Two of the shortest, at eight letters, are caesious and anemious (OED); and aerious (OED) has only seven letters. Some others are abstentious, acheilous, arsenious, arterious, tragedious, fracedinous, and Gadsprecious (all in OED). Considering y as a vowel, the suffix -ly can be added to a number of these words; thus the shortest word containing six unique vowels in alphabetical order is aeriously, with nine letters (OED); the much more common facetiously has eleven letters.
    One does not mess with regular expressions.
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