the total number of arrangements which can be made out of the letters of the word ALGEBRA without altering the relative position of the vowels and consonants is

Options

A. `(7!)/(2!)`
B. `7/(2! 5!)`
C. `4!3!`
D. `(4!3!)/2`

Answer

To solve the problem of finding the total 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, E, A (3 vowels)
- Consonants: L, G, B, R (4 consonants)

### Step 2: Determine the Arrangement of Vowels
Since the vowels must maintain their relative positions, we can treat them as a single group. The arrangement of the vowels A, E, A can be calculated using the formula for permutations of multiset:
\[ \text{Number of arrangements of vowels} = \frac{n!}{p_1! \times p_2!} \]
where \( n \) is the total number of vowels, and \( p_1, p_2, \ldots \) are the frequencies of the repeated vowels.

Here, we have:
- Total vowels = 3 (A, E, A)
- A appears 2 times, E appears 1 time.

Thus, the number of arrangements of the vowels is:
\[ \text{Arrangements of vowels} = \frac{3!}{2! \times 1!} = \frac{6}{2} = 3 \]

### Step 3: Determine the Arrangement of Consonants
The consonants L, G, B, R can be arranged freely among themselves. The number of arrangements of the 4 consonants is given by:
\[ \text{Arrangements of consonants} = 4! = 24 \]

### 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 to get the total arrangements:
\[ \text{Total arrangements} = \text{Arrangements of vowels} \times \text{Arrangements of consonants} = 3 \times 24 = 72 \]

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

---