Multiple choice

In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

  1. 6!/2!

  2. 3!*3!

  3. (3!*3!)/2!

  4. (4!*3!)/2!

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

The word ABACUS has 6 letters: A, B, A, C, U, S. Vowels are A, A, U. Treat (AAU) as one unit. We have 4 units: (AAU), B, C, S. These can be arranged in 4! ways. Within the unit (AAU), the letters can be arranged in 3!/2! ways. Total = 4! * (3!/2!) = 24 * 3 = 72. Option D matches this expression.

AI explanation

The word ABACUS has 6 letters where A is repeated twice, and it contains three vowels. Treating the three vowels as a single unit, we arrange this unit alongside the three consonants, which can be done in 4! ways. The three vowels can be arranged among themselves in 3! ways, making the total arrangements (4! x 3!)/2!, where 2! corrects for the identical vowels A.

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