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

Answer

To find the number of ways to arrange the letters of the word "ABACUS" such that the vowels always appear together, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Vowels and Consonants**:
The word "ABACUS" contains the vowels A, A, and U. The consonants are B, C, and S.

2. **Group the Vowels Together**:
Since the vowels must always appear together, we can treat the group of vowels (AAU) as a single unit. Thus, we can represent it as V (where V = AAU).

3. **Count the Total Units**:
Now, we have the following units to arrange:
- V (the group of vowels)
- B (consonant)
- C (consonant)
- S (consonant)

This gives us a total of 4 units to arrange: V, B, C, and S.

4. **Calculate the Arrangements of the Units**:
The number of ways to arrange these 4 units is given by 4! (factorial of 4):
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

5. **Arrange the Vowels within the Group**:
Next, we need to arrange the vowels within the group V (AAU). The vowels consist of two A's and one U. The number of ways to arrange these vowels is given by:
\[ \frac{3!}{2!} = \frac{3 \times 2 \times 1}{2 \times 1} = 3 \]
(Here, 3! accounts for the total arrangements of 3 letters, and we divide by 2! to account for the repetition of the letter A.)

6. **Calculate the Total Arrangements**:
Finally, we multiply the number of arrangements of the units by the arrangements of the vowels:
\[ \text{Total arrangements} = 4! \times \frac{3!}{2!} = 24 \times 3 = 72 \]

### Final Answer:
Thus, the total number of ways to arrange the letters of the word "ABACUS" such that the vowels always appear together is **72**.