Multiple choice

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. 34,000

  2. 37,000

  3. 36,000

  4. 35,000
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total permutations of DAUGHTER (8 letters, all distinct) = 8! = 40320. Vowels are A, U, E (3). Treat (AUE) as one unit. Permutations with vowels together = 6! * 3! = 720 * 6 = 4320. Vowels never together = 40320 - 4320 = 36000.

AI explanation

Using the complementary method, first find the total unrestricted arrangements of the 8 distinct letters, which is 8! = 40320. Next, treat the 3 vowels as a single unit to find the arrangements where they are together, giving 6! arrangements for the units and 3! internal arrangements for the vowels, resulting in 720 * 6 = 4320 ways. Subtracting the together arrangements from the total gives 40320 - 4320 = 36000 ways where the vowels never come together.

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