Question

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

• A. \(\frac{6!}{2!}\)
• B. \(3! \ast 3!\)
• C. \(\frac{(3! \ast 3!)}{2!}\)
• D. \(\frac{(4! \ast 3!)}{2!}\)

Answer 1

The Correct Option is D

Given the word ABACUS, we need to rearrange the letters such that the vowels always appear together.
1. Identify the vowels: A, A, U
2. Identify the consonants: B, C, S
We can consider the group of vowels (AAU) as a single unit. So, we have the following units to arrange:
- (AAU), B, C, S
These four units can be arranged in \(4!\) ways.
Within the group (AAU), the vowels can be arranged in \(\frac{3!}{2!}\) ways since there are two A's which are identical.
Therefore, the total number of arrangements is:
\[4! \times \frac{3!}{2!} = 24 \times 3 = 72\]
Hence, the number of ways to rearrange the letters such that the vowels always appear together is \(\frac{(4! \ast 3!)}{2!}\).
Correct Answer Option: D \(\frac{(4! \ast 3!)}{2!}\).