Find the number of arrangements of the letters of the word 'ALGEBRA' without altering the relative positions of the vowels and the consonants.

Explanation

To solve the problem of finding the number of arrangements of the letters in the word "ALGEBRA" without altering the relative positions of the vowels and consonants, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "ALGEBRA" consists of the following letters:
- Vowels: A, A, E (total of 3 vowels)
- Consonants: L, G, B, R (total of 4 consonants)

### Step 2: Calculate the arrangements of vowels
We have 3 vowels: A, A, E. Since the letter A is repeated, we use the formula for permutations of multiset:

\[ \text{Arrangements of vowels} = \frac{n!}{p_1! \times p_2!} \]

Where:
- \( n \) is the total number of vowels,
- \( p_1 \) is the factorial of the count of the first repeated letter (A),
- \( p_2 \) is the factorial of the count of the second letter (E).

For our vowels:
- Total vowels, \( n = 3 \) (A, A, E)
- Repeated A's, \( p_1 = 2! \)
- E's, \( p_2 = 1! \)

So, the calculation is:

\[ \text{Arrangements of vowels} = \frac{3!}{2! \times 1!} = \frac{6}{2 \times 1} = 3 \]

### Step 3: Calculate the arrangements of consonants
We have 4 consonants: L, G, B, R. All consonants are distinct, so we can simply calculate the factorial of the number of consonants:

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

### Step 4: Combine the arrangements
Since the arrangements of vowels and consonants are independent, we can multiply the number of arrangements of vowels by the number of arrangements of consonants:

\[ \text{Total arrangements} = \text{Arrangements of vowels} \times \text{Arrangements of consonants} = 3 \times 24 = 72 \]

### Final Answer
The total number of arrangements of the letters of the word "ALGEBRA" without altering the relative positions of the vowels and consonants is **72**.

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