How many words can be formed with the letters of the word 'GUJRAT', if all vowels are always together?

Answer

To solve the problem of how many words can be formed with the letters of the word 'GUJRAT' if all vowels are always together, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word 'GUJRAT' consists of the letters: G, U, J, R, A, T.
Here, the vowels are U and A, and the consonants are G, J, R, T.

### Step 2: Treat the vowels as a single unit
Since we want the vowels to always be together, we can treat the combination of the vowels (UA) as a single unit. Therefore, we can represent the letters as:
- (UA), G, J, R, T

This gives us a total of 5 units to arrange: (UA), G, J, R, T.

### Step 3: Calculate the arrangements of the units
The number of ways to arrange these 5 units is given by the factorial of the number of units:
\[ 5! = 120 \]

### Step 4: Calculate the arrangements of the vowels within their unit
Within the unit (UA), the vowels can be arranged among themselves. The arrangements of the vowels U and A can be calculated as:
\[ 2! = 2 \]

### Step 5: Calculate the total arrangements
To find the total number of arrangements where the vowels are always together, we multiply the arrangements of the units by the arrangements of the vowels:
\[ \text{Total arrangements} = 5! \times 2! = 120 \times 2 = 240 \]

### Final Answer
Thus, the total number of words that can be formed with the letters of the word 'GUJRAT', with all vowels together, is **240**.

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