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

Answer: (D)

$\frac{(4! \ast 3!)}{2!}$

Answer 2

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 AnswerOption: D $\frac{(4! \ast 3!)}{2!}$.