Question

Find the number of different $8$-letter arrangements that can be made from the letters of the word DAUGHTER so that all vowels do not occur together.

Answer 1

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 $6!\times 3!=4320$

Total no of ways in which it can be arranged so that the vowels are not all together = $8!-6!\times 3!=40320-4320=36000$