The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is :
Solutions · 1
Official solution
DAUGHTER has 8 distinct letters, with 3 vowels (A, U, E).
Total arrangements =8!=40320.
Vowels together: treat 3 vowels as one block, arrange 6 units in 6! ways, vowels within block in 3! ways: 6!×3!=4320.
\[\text{Required} = 8! - 6!\times3! = 40320 - 4320 = 36000\]