Number of words that can be formed with the letters of the word ALGEBRA so that all the vowels are seperated (or no two vowals come together) is

Options

A. 720
B. 2160
C. 1440
D. 1200

Answer

To solve the problem of finding the number of words that can be formed with the letters of the word "ALGEBRA" such that all the vowels are separated, we can follow these steps:

### Step 1: Identify the letters
The word "ALGEBRA" consists of 7 letters: A, L, G, E, B, R, A.

- **Vowels**: A, A, E (3 vowels)
- **Consonants**: L, G, B, R (4 consonants)

### Step 2: Arrange the consonants
First, we need to arrange the consonants. The consonants are L, G, B, and R. The number of ways to arrange these 4 consonants is given by:

\[ 4! = 24 \]

### Step 3: Determine the positions for the vowels
When the 4 consonants are arranged, they create gaps where the vowels can be placed. For example, if the consonants are arranged as C1, C2, C3, C4, the possible positions for the vowels are:

- _ C1 _ C2 _ C3 _ C4 _

This gives us 5 possible positions (gaps) for the vowels.

### Step 4: Select positions for the vowels
We need to choose 3 out of these 5 positions to place the vowels. The number of ways to choose 3 positions from 5 is given by:

\[ \binom{5}{3} = 10 \]

### Step 5: Arrange the vowels
Now, we need to arrange the vowels A, A, E in the selected positions. The number of arrangements of the vowels, considering that A is repeated, is given by:

\[ \frac{3!}{2!} = 3 \]

### Step 6: Calculate the total arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to arrange the consonants, the number of ways to choose positions for the vowels, and the number of ways to arrange the vowels:

\[ \text{Total arrangements} = (4!) \times \left(\binom{5}{3}\right) \times \left(\frac{3!}{2!}\right) \]

Substituting the values we calculated:

\[ \text{Total arrangements} = 24 \times 10 \times 3 = 720 \]

### Final Answer
The total number of words that can be formed with the letters of the word "ALGEBRA" such that all the vowels are separated is **720**.

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