The number of arrangements of the letters the word "INDEPENDENCE" in which all the vowels always occur together is .

Options

A. `18000`
B. `14800`
C. `33600`
D. `16800`

Answer

To find the number of arrangements of the letters in the word "INDEPENDENCE" such that all the vowels always occur together, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "INDEPENDENCE" consists of the following letters:
- Vowels: I, E, E, E (4 vowels)
- Consonants: N, D, P, N, D, C (6 consonants)

### Step 2: Treat all vowels as a single unit
Since we want all the vowels to occur together, we can treat the group of vowels (IEEE) as a single unit. Thus, the arrangement can be visualized as:
- (IEEE), N, D, P, N, D, C

This gives us a total of 7 units to arrange: (IEEE), N, D, P, N, D, C.

### Step 3: Calculate arrangements of the units
The number of arrangements of these 7 units is calculated using the formula for permutations of multiset:
\[ \text{Arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots} \]
where \(n\) is the total number of units, and \(p_1, p_2, \ldots\) are the counts of each indistinguishable unit.

In our case:
- Total units = 7
- The consonants N and D are repeated (N appears 2 times and D appears 2 times).

Thus, the number of arrangements is:
\[ \text{Arrangements} = \frac{7!}{2! \times 2!} \]

### Step 4: Calculate \(7!\) and the factorials
Calculating \(7!\):
\[ 7! = 5040 \]
Calculating \(2!\):
\[ 2! = 2 \]

### Step 5: Substitute and simplify
Now substituting back:
\[ \text{Arrangements} = \frac{5040}{2 \times 2} = \frac{5040}{4} = 1260 \]

### Step 6: Arrange the vowels
Next, we need to arrange the vowels within their group (IEEE). The vowels I, E, E, E have repetitions as well:
\[ \text{Arrangements of vowels} = \frac{4!}{1! \times 3!} = \frac{24}{6} = 4 \]

### Step 7: Total arrangements
Finally, the total arrangements where all vowels are together is:
\[ \text{Total arrangements} = \text{Arrangements of units} \times \text{Arrangements of vowels} = 1260 \times 4 = 5040 \]

Thus, the number of arrangements of the letters in the word "INDEPENDENCE" in which all the vowels always occur together is **5040**.