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How many distinct ways can the letters of the word UNDETERMINED be arranged so that all the vowels are in alphabetical order? Do I need to consider the positions of the vowels?

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Hint: This is the same as finding the number of arrangements of the word XNDXTXRMXNXD. Given any arrangment of this word, we can just put the vowels in alphabetical order wherever X's appear.

So, for example, the arrangement XXXNDTDRMNXX of XNDXTXRMXNXD corresponds only to the actual arrangement EEENDTDRMNIU of the original word.

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For the word UNDETERMINED, vowels must be in the order "EEEIU"

If we ignore the vowels for a minute here and replace them with a dash, we have to arrange the following: -ND-T-RM-N-D. We can do that as we don't care of the rearrangements within the dashes as only one possibility is there ("EEEIU")

So the possible ways are simply:

$${12! \over (5! * 2! * 2!)}=997920$$

12! for arranging all letters

5! ways removed for the vowels ( since only 1 way is possible )

2! ways each removed for the "N"s and "D"s

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