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      Question

      How many unique arrangements can be made using all the

      letters of the word "DELUSION", ensuring that the vowels do not appear together?
      A 32580 Correct Answer Incorrect Answer
      B 42420 Correct Answer Incorrect Answer
      C 37440 Correct Answer Incorrect Answer
      D 36240 Correct Answer Incorrect Answer
      E 38560 Correct Answer Incorrect Answer

      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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