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Q.

Find the number of different 8− letter arrangements that can be made from the letters of the word “DAUGHTER” such that all vowels do not occur together.

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a

36000

b

4320

c

10080

d

40230

answer is A.

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Detailed Solution

Total number of letters in DAUGHTER = 8
Total number of permutations of 8 letters = 8P8 = 40320
And vowels in DAUGHTER=A, V and E.
Since all vowels occur together, assume AUE as a single object and left letters are DGHTR
Arrange 3 vowels = 3P3 = 720
Arrange 6 letters = 6P6 = 4320
All vowels do not occur together =total number of permutations – number of permutations of all vowels occurring together =40320 – 4320 = 36000
Hence, Option (1) is the correct answer.

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Find the number of different 8− letter arrangements that can be made from the letters of the word “DAUGHTER” such that all vowels do not occur together.