Question

Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that

1. all vowels occur together.

2. all vowels do not occur together.

Answer 1

1. DAUGHTER this word has 8 different letters. A, U, E are the vowels. Treat these 3 as one unit, then there are 6 units and can be permuted in 6! ways. The above vowels can be permuted in 3! ways. 

Hence the total number of words is 6! × 3! = 4320.

2. Number of words all vowels do not occur together = Total number of different words – number of words in which vowels come together

= 8! – 6! × 3! 

= 6!(8 × 7 – 6) 

= 2 × 6!(28 – 3)

= 50 × 6! = 36000.