Question

Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that

1) all vowels occur together

2) all vowels do not occur together.

Answer 1

Total number of words formed from the word DAUGHTER = 8! = 40320 words

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(1) There are 3 vowels in DAUGHTER - A,U,E.... If vowels are tied together and used as single letter then:-

(AUE)DGHTR

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Here the brackets represent them to be together. Hence now the no. of words formed will be = 6! * 3!, since AUE can be arranged themselves in 3!.

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So if all vowels occur together then the no. of words formed = 6! * 3! = 720 * 6 = 4320 words .

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(2) The no. of words formed if vowels do not occur together = Total no. of words formed from DAUGHTER - No. of words formed when vowels occur together.

Henceno. of words formed if vowels do not occur together = 8! - (6!*3!) = 40320 - 4320 = 36000 words.

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Hence answers:-

(1) 4320 words

(2) 36000 words