Question

The number of words that can be formed using all the letters of the word "DAUGHTER" such that all the vowels never come together, is:

• A. \( 34000 \)
• B. \( 37000 \)
• C. \( 36000 \)
• D. \( 35000 \)

Hint:

When counting arrangements where certain items must not be together, first count the total arrangements, then subtract the unwanted cases (where the items are together).

Answer 1

The Correct Option is C

Step 1: Total number of words. The total number of words that can be formed using all the letters of the word "DAUGHTER" is: \[ \text{Total words} = 8! = 40320. \] Step 2: Words with all vowels together. The vowels in "DAUGHTER" are A, U, and E. If we treat these vowels as a single entity, the number of arrangements of the letters is: \[ \text{Words with vowels together} = 6! \times 3! = 720 \times 6 = 4320. \] Step 3: Words with vowels not together. To find the number of words where the vowels are not together, subtract the number of words with vowels together from the total number of words: \[ \text{Words with vowels not together} = 8! - 6! \times 3! = 40320 - 4320 = 36000. \]