In how many ways can the letters of the word “VALEDICTORY” be arranged so that the vowels are never separated?

Options

A. 883490
B. 967680
C. 563680
D. 483840

Answer

To solve the problem of arranging the letters of the word "VALEDICTORY" such that the vowels are never separated, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "VALEDICTORY" consists of the following letters:
- Vowels: A, E, I, O (total of 4 vowels)
- Consonants: V, L, D, C, T, R, Y (total of 7 consonants)

### Step 2: Treat the vowels as a single unit
Since we want the vowels to always be together, we can treat the group of vowels (AEIO) as a single unit. This means we now have:
- 1 unit of vowels (AEIO)
- 7 consonants (V, L, D, C, T, R, Y)

### Step 3: Count the total units
Now, we have a total of 8 units to arrange:
- 1 unit of vowels + 7 consonants = 8 units

### Step 4: Arrange the units
The number of ways to arrange these 8 units is given by the factorial of the number of units:
\[ 8! \]

### Step 5: Arrange the vowels within their unit
Within the vowel unit (AEIO), the 4 vowels can be arranged among themselves. The number of ways to arrange these 4 vowels is:
\[ 4! \]

### Step 6: Calculate the total arrangements
To find the total number of arrangements where the vowels are together, we multiply the arrangements of the units by the arrangements of the vowels:
\[ \text{Total arrangements} = 8! \times 4! \]

### Step 7: Compute the values
Now, we compute the factorials:
- \( 8! = 40320 \)
- \( 4! = 24 \)

Thus, the total arrangements are:
\[ 40320 \times 24 = 967680 \]

### Final Answer
The total number of ways to arrange the letters of the word "VALEDICTORY" such that the vowels are never separated is **967680**.

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