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

Solution: (a)Number of 8 different arrangements with all vowels occur together-
Vowels include-(a,u,e) so mark them as 1
now we have 6 things to arrange , that can be done in $6!\times 3!=4320$ Mulitplying by 3! for internal arrangements of vowels.
(b) all vowels do not occur together
=>Total arrangements-(when vowels occur together)
=>$8!-4320=36000$