find the number of arrangements which can be made out of the letters of the word 'algebra', without altering the relative positions of vowels and consonants.

Options

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

Answer

To solve the problem of finding the number of arrangements that can be made out of the letters of the word "algebra" without altering the relative positions of vowels and consonants, we can follow these steps:

### Step 1: Identify the vowels and consonants
In the word "algebra", the letters are:
- Vowels: a, e, a (3 vowels: a appears twice)
- Consonants: l, g, b, r (4 consonants)

### Step 2: Calculate the arrangements of the vowels
The formula for arranging n items where some items are identical is given by:
\[ \frac{n!}{p_1! \cdot p_2! \cdots p_k!} \]
where \( n \) is the total number of items, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the identical items.

For the vowels (a, e, a):
- Total vowels = 3 (a, e, a)
- The letter 'a' appears twice.

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

### Step 3: Calculate the arrangements of the consonants
For the consonants (l, g, b, r):
- Total consonants = 4 (l, g, b, r)
- All consonants are different.

Thus, the number of arrangements of the consonants is:
\[ \text{Arrangements of consonants} = 4! = 24 \]

### Step 4: Calculate the total arrangements
Since the arrangements of vowels and consonants are independent, we 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 in the word "algebra" without altering the relative positions of vowels and consonants is **72**.

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