1.

Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that1. all vowels occur together.2. all vowels do not occur together.

Answer»

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.



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