Question

The total number of ways the word 'DAUGHTER' can be arranged so that all vowels don't occur together:

• A. 36000
• B. 37000
• C. 35000
• D. 38000

Hint:

When arranging words with restrictions, first calculate the number of ways to arrange the consonants and then the number of ways to arrange the vowels in available positions.

Answer 1

The Correct Option is C

The word "DAUGHTER" has 8 letters, with 3 vowels (A, U, E) and 5 consonants (D, G, H, T, R). To arrange the letters so that the vowels don't occur together: - First, arrange the 5 consonants: \( 5! = 120 \) ways. - Now, place the 3 vowels in the 6 available positions (before the first consonant, between consonants, and after the last consonant). The number of ways to choose 3 positions from 6 is \( \binom{6}{3} = 20 \). - The 3 vowels can be arranged in these positions in \( 3! = 6 \) ways. Thus, the total number of arrangements is \( 5! \times \binom{6}{3} \times 3! = 120 \times 20 \times 6 = 36000 \). Thus, the correct answer is option (1).