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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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