Multiple choice

In how many different ways can the letters of the word \"MACHINE\" be arranged such that the vowels occupy only the odd positions?

  1. 144

  2. 5040

  3. 348

  4. 576

  5. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

MACHINE has 7 letters: 3 vowels (A, I, E) and 4 consonants (M, C, H, N). Vowels must occupy odd positions (1, 3, 5, 7). There are 4 odd positions and 3 vowels, so 4P3 = 24 ways. The 4 consonants occupy the remaining 4 positions in 4! = 24 ways. Total = 24 * 24 = 576.

AI explanation

The word MACHINE has 7 letters with 3 vowels (A, I, E) and 4 consonants (M, C, H, N). The total number of letters is 7, so there are exactly 4 odd positions and 3 even positions. Since the vowels must occupy only odd positions, the 3 vowels can be arranged in the 4 odd positions in 4 permutation 3 ways, equalling 24. The 4 consonants can then be arranged in the remaining 4 positions in 4 factorial ways, equalling 24. The total number of arrangements is 24 times 24, which equals 576.

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