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:
Step-by-Step Solution
Key Concept: Total arrangements minus arrangements where all $3$ vowels are clumped as one block.
DAUGHTER has $8$ distinct letters; vowels $\{A,U,E\}$, consonants $\{D,G,H,T,R\}$.
Total $=8!=40320.$
All vowels together: treat $\{A,U,E\}$ as one block. Arrange $6$ items: $6!$; internal arrangement: $3!$. Total $=6!\cdot 3!=720\cdot 6=4320.$
Vowels never all together $=40320-4320=36000.$
Correct Answer: 1