The number of words that can be be formed out of the letters of the word 'INDEPENDENT', so that vowels and consonants occur together is

Options

A. 960
B. 3360
C. 2160
D. 4320

Answer

To solve the problem of finding the number of words that can be formed from the letters of the word "INDEPENDENT" such that the vowels and consonants occur together, we can follow these steps:

### Step 1: Identify the Vowels and Consonants
The word "INDEPENDENT" consists of the following letters:
- Vowels: I, E, E, E (total 4 vowels, with E repeated 3 times)
- Consonants: N, N, N, D, D, P, T (total 7 consonants, with N repeated 3 times and D repeated 2 times)

### Step 2: Treat Vowels and Consonants as Single Entities
Since we want the vowels and consonants to occur together, we can treat all vowels as one single entity and all consonants as another single entity. Thus, we have:
- One entity for vowels (V)
- One entity for consonants (C)

### Step 3: Arrange the Entities
Now, we can arrange these two entities (V and C). The number of ways to arrange 2 entities is:
\[ 2! = 2 \]

### Step 4: Arrange the Vowels
Next, we need to arrange the vowels within their entity. The vowels are I, E, E, E. The number of arrangements of these vowels is given by the formula for permutations of multiset:
\[ \frac{n!}{n_1! \times n_2! \times \ldots} \]
where \( n \) is the total number of items to arrange, and \( n_1, n_2, \ldots \) are the counts of each repeated item.

For the vowels:
- Total vowels = 4 (I, E, E, E)
- E is repeated 3 times.

Thus, the number of arrangements of the vowels is:
\[ \frac{4!}{3!} = \frac{24}{6} = 4 \]

### Step 5: Arrange the Consonants
Now, we arrange the consonants N, N, N, D, D, P, T. The number of arrangements of these consonants is:
- Total consonants = 7 (N, N, N, D, D, P, T)
- N is repeated 3 times and D is repeated 2 times.

Thus, the number of arrangements of the consonants is:
\[ \frac{7!}{3! \times 2!} = \frac{5040}{6 \times 2} = \frac{5040}{12} = 420 \]

### Step 6: Calculate the Total Arrangements
Finally, we multiply the arrangements of the entities, vowels, and consonants together:
\[ \text{Total arrangements} = (2!) \times (\text{arrangements of vowels}) \times (\text{arrangements of consonants}) \]
\[ = 2 \times 4 \times 420 = 3360 \]

### Conclusion
The total number of words that can be formed from the letters of the word "INDEPENDENT" such that the vowels and consonants occur together is **3360**.

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