Find the number of different $8$ -letter arrangements that can be made from the letters of the word DAUGHTER so that all vowels occur together.

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Solution

Total no of letters $= 8$
Consonants $= 5$
Vowels $= 3$
Treating all the vowels as a single block and they can arrage themselves in $3!$ ways.
Hence, total no of ways all the letters can be arranged is $6! \times 3! = 4320$

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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.

Find the no.of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that:

(a)All vowels occur together.

(b)All vowels do not occur together.

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