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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. |
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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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