The number of words formed from all the letters of DAUGHTER such that all vowels are never together is
Step-by-Step Solution
Key Concept: Use complementary counting: subtract arrangements where all 3 vowels (A, U, E) are treated as a single block from the total $8!$ arrangements.
Total arrangements: $8!=40320$.
All 3 vowels (A, U, E) together: treat as one block; $6!$ arrangements of the 6 units $\times\,3!$ internal arrangements of vowels $=720\times6=4320$.
Vowels never all together: $40320-4320=36000$.
Correct Answer: 1