Multiple choice

The number of ways the letters of the word MATHEMATICS be permuted so that the vowels never comes together

  1. $ \displaystyle \frac{11!}{2!2!2!} $
  2. $ \displaystyle \frac{8!4!}{2!2!2!} $
  3. $ \displaystyle \frac{7!5!}{2!2!2!} $

  4. None of these
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D Correct answer
AI explanation

The total permutations of the 11 letters in MATHEMATICS, which contains repeating M's, A's, and T's, is 11!/(2!2!2!). To find the permutations where the 4 vowels (A, A, E, I) do not come together, we first group them into a single block. The block and the 7 consonants can be arranged in 8!/(2!2!) ways, and the vowels within the block can be arranged in 4!/2! ways. The number of permutations where vowels are together is (8! x 4!)/(2!2!2!), so subtracting this from the total permutations gives (11! - 8!4!)/(2!2!2!), which does not match any of the provided expressions.

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