Find number of othe ways in which word 'KOLAVARI' can be arranged, if all vowels are separate from each other ?

Options

A. 289
B. 2880
C. 1439
D. 1440

Answer

To find the number of ways to arrange the word "KOLAVARI" such that all vowels are separated from each other, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "KOLAVARI" consists of the following letters:
- Vowels: O, A, A, I (4 vowels)
- Consonants: K, L, V, R (4 consonants)

### Step 2: Arrange the consonants
First, we will arrange the consonants. The consonants are K, L, V, R. The number of ways to arrange these 4 consonants is given by:
\[ 4! = 24 \]

### Step 3: Create spaces for the vowels
Once the consonants are arranged, we need to create spaces for the vowels. Arranging the 4 consonants creates 5 potential spaces for the vowels (before the first consonant, between the consonants, and after the last consonant). For example, if we arrange the consonants as K L V R, the spaces would look like this:
\[ \_ K \_ L \_ V \_ R \_ \]

### Step 4: Choose spaces for the vowels
We need to select 4 out of these 5 spaces to place the vowels. The number of ways to choose 4 spaces from 5 is given by:
\[ \binom{5}{4} = 5 \]

### Step 5: Arrange the vowels
Now, we need to arrange the vowels O, A, A, I. Since the letter A is repeated, the number of distinct arrangements of these vowels is given by:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

### Step 6: Calculate the total arrangements
Now, we can calculate the total number of arrangements by multiplying the number of arrangements of consonants, the number of ways to choose spaces, and the arrangements of vowels:
\[ \text{Total arrangements} = (4!) \times \binom{5}{4} \times \frac{4!}{2!} \]
Substituting the values we calculated:
\[ \text{Total arrangements} = 24 \times 5 \times 12 = 1440 \]

### Final Answer
Thus, the total number of ways to arrange the word "KOLAVARI" such that all vowels are separated from each other is **1440**.