Question
How many unique arrangements can be made using all the
letters of the word "DELUSION", ensuring that the vowels do not appear together?Solution
If we take all the vowels to be a single letter, then
Total number of letters = 5 [EUIO is taken as a single letter]
Number of ways of arranging with all the vowels together = 5! × 4! = 120 × 24 = 2880
Number of ways of arranging without any condition = 8! = 40320
So, number of ways the word can be arranged so that all the vowels never occur together = 40320 – 2880 = 37440
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