Question ID: #238
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
- (1) 34000
- (2) 37000
- (3) 36000
- (4) 35000
Solution:
Word: DAUGHTER
Total letters = 8.
Vowels: A, U, E (3 letters).
Consonants: D, G, H, T, R (5 letters).
Total arrangements
Total words $= 8! = 40,320$.
Arrangements where vowels are together
Treat (AUE) as one unit.
Total entities = 5 (consonants) + 1 (vowel group) = 6.
Arrangements $= 6! \times 3!$ (internal arrangement of vowels).
$= 720 \times 6 = 4,320$.
Vowels never together
Required = Total $-$ Together
$= 40,320 – 4,320 = 36,000$.
Ans. (3)
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