Multiple choice

In how many ways can the letters of the word MANAGEMENT be arranged such that no two vowels appear together?

  1. 37,800

  2. 75,600

  3. 25,200

  4. 21,600
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A Correct answer
Explanation

MANAGEMENT has 10 letters: M, A, N, A, G, E, M, E, N, T. Vowels: A, A, E, E (4). Consonants: M, N, G, M, N, T (6). Arrange consonants: 6! / (2! * 2!) = 720 / 4 = 180. Spaces for vowels: _ C _ C _ C _ C _ C _ C _ (7 spaces). Choose 4 spaces: 7C4 = 35. Arrange vowels: 4! / (2! * 2!) = 6. Total = 180 * 35 * 6 = 37800.

AI explanation

The word MANAGEMENT has 10 letters, consisting of 4 vowels (A, A, E, E) and 6 consonants (M, M, N, N, G, T). We first arrange the 6 consonants, which can be done in 6! / (2! * 2!) = 180 ways, creating 7 gaps around them. To ensure no two vowels are together, we place the 4 vowels in these 7 gaps, which can be done in 7C4 * 4! / (2! * 2!) = 105 ways. Multiplying the arrangements of consonants and vowels gives 180 * 105, resulting in 37,800 total arrangements.

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