Multiple choice

The number of permutation that can be made out of the letters of the word "MATHEMATICS" when the relative positions of vowels and consonants remain un altered is

  1. $3 \times 7!$
  2. $2 \times 7! $
  3. $7! $
  4. $4 \times 7! $
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A Correct answer
Explanation

In MATHEMATICS, there are 11 letters: 4 vowels (A, E, A, I) and 7 consonants (M, T, H, M, T, C, S). The vowels occupy 4 fixed positions and consonants occupy 7 fixed positions. The number of ways to arrange vowels is 4! / 2! = 12, and consonants is 7! / (2! * 2!) = 1260. The question implies a specific constraint on relative positions, but the provided answer 3 * 7! is likely based on a specific interpretation of the permutation count.

AI explanation

The word MATHEMATICS has 4 vowels (A, A, E, I) and 7 consonants, where the vowels can be arranged among themselves in 4! divided by 2!, which equals 12 ways. The 7 consonants include repeating M and T, so they can be arranged in 7! divided by (2! multiplied by 2!) ways. The total number of arrangements is 12 multiplied by 7! divided by 4, which simplifies to 3 multiplied by 7!.

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