How many different letter arrangements can be made from the letter of the word EXTRA in such a way that the vowels are always together?

Options

A. 48
B. 60
C. 40
D. 30

Answer

To solve the problem of how many different letter arrangements can be made from the letters of the word "EXTRA" such that the vowels are always together, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Letters**: The word "EXTRA" consists of the letters E, X, T, R, A. Here, we have:
- Vowels: E, A
- Consonants: X, T, R

2. **Group the Vowels Together**: Since we want the vowels (E and A) to always be together, we can treat them as a single unit or block. We can represent this block as (EA).

3. **Count the Units**: Now, we can list the units we have:
- The block (EA)
- The consonants X, T, R

This gives us a total of 4 units to arrange: (EA), X, T, R.

4. **Calculate Arrangements of the Units**: The number of ways to arrange these 4 units is given by the factorial of the number of units:
\[ \text{Arrangements of units} = 4! = 24 \]

5. **Arrange the Vowels Within the Block**: The vowels E and A within the block (EA) can be arranged among themselves. The number of ways to arrange 2 vowels is:
\[ \text{Arrangements of vowels} = 2! = 2 \]

6. **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:
\[ \text{Total arrangements} = 4! \times 2! = 24 \times 2 = 48 \]

### Final Answer:
The total number of different letter arrangements of the word "EXTRA" such that the vowels are always together is **48**.

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