What is the number of words that can be formed from the letters of the word 'UNIVERSAL', the vowels remaining always together ?

Options

A. 720
B. 1440
C. 17280
D. 21540

Explanation

To solve the problem of finding the number of words that can be formed from the letters of the word 'UNIVERSAL' with the condition that the vowels always remain together, we can follow these steps:

### Step 1: Identify the Vowels and Consonants
The word 'UNIVERSAL' consists of the following letters:
- Vowels: U, I, E, A (4 vowels)
- Consonants: N, V, R, S, L (5 consonants)

### Step 2: Treat Vowels as a Single Entity
Since the vowels must always be together, we can treat them as a single entity or block. This means we can represent the vowels as a single letter, say 'V'. Thus, we now have the following letters to arrange:
- V (the block of vowels)
- N
- V
- R
- S
- L

This gives us a total of 6 entities to arrange: V, N, R, S, L.

### Step 3: Calculate the Arrangements of the Entities
The number of ways to arrange these 6 entities is given by the factorial of the number of entities:
\[ 6! = 720 \]

### Step 4: Arrange the Vowels Within Their Block
Next, we need to arrange the vowels within their block. The vowels U, I, E, A can be arranged among themselves. The number of arrangements of the 4 vowels is:
\[ 4! = 24 \]

### Step 5: Calculate the Total Arrangements
To find the total number of arrangements of the letters in 'UNIVERSAL' with the vowels together, we multiply the number of arrangements of the entities by the number of arrangements of the vowels:
\[ \text{Total arrangements} = 6! \times 4! = 720 \times 24 = 17280 \]

### Final Answer
Thus, the total number of words that can be formed from the letters of the word 'UNIVERSAL' with the vowels remaining always together is **17280**.

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