# Thread: Arranging sequences of letters in a word

1. ## 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?

2. Yes those are both correct.

3. More trivia . . .

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

Another is FACETIOUS.

Do you know the others?

4. Originally Posted by uberbandgeek6
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.

5. 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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### a word arranging aeiou

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