In how many ways the letters of the word "ARRANGE" can be arranged without altering the relative positions of vowels & consonants?

Answer

To solve the problem of arranging the letters of the word "ARRANGE" without altering the relative positions of vowels and consonants, we can follow these steps:

### Step 1: Identify Vowels and Consonants
In the word "ARRANGE", we identify the vowels and consonants:
- Vowels: A, A, E (3 vowels)
- Consonants: R, R, N, G (4 consonants)

### Step 2: Arrange the Consonants
The consonants R, R, N, G can be arranged among themselves. Since the letter R repeats, we use the formula for permutations of multiset:

\[ \text{Arrangements of consonants} = \frac{n!}{p_1! \cdot p_2! \cdots} \]

Where:
- \( n \) is the total number of consonants,
- \( p_1, p_2, \ldots \) are the frequencies of the repeating consonants.

For the consonants R, R, N, G:
- Total consonants \( n = 4 \)
- R repeats 2 times.

Thus, the arrangements of consonants is:

\[ \text{Arrangements of consonants} = \frac{4!}{2!} = \frac{24}{2} = 12 \]

### Step 3: Arrange the Vowels
Next, we arrange the vowels A, A, E. Again, since A repeats, we apply the same formula:

For the vowels A, A, E:
- Total vowels \( n = 3 \)
- A repeats 2 times.

Thus, the arrangements of vowels is:

\[ \text{Arrangements of vowels} = \frac{3!}{2!} = \frac{6}{2} = 3 \]

### Step 4: Calculate Total Arrangements
Now, we find the total arrangements by multiplying the arrangements of consonants and vowels:

\[ \text{Total arrangements} = \text{Arrangements of consonants} \times \text{Arrangements of vowels} = 12 \times 3 = 36 \]

### Final Answer
Therefore, the total number of ways to arrange the letters of the word "ARRANGE" without altering the relative positions of vowels and consonants is **36**.

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