How many words can be formed by arranging the letters of the word 'INDIA', so that the vowels are never together?

Explanation

To solve the problem of how many words can be formed by arranging the letters of the word 'INDIA' such that the vowels are never together, we can follow these steps:

### Step 1: Identify the letters and their frequencies
The word 'INDIA' consists of 5 letters: I, N, D, I, A.
- Vowels: I, I, A (3 vowels, with 'I' repeated twice)
- Consonants: N, D (2 consonants)

### Step 2: Calculate the total arrangements of the letters
To find the total arrangements of the letters in 'INDIA', we use the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots} \]

Where \( n \) is the total number of letters and \( p_i \) are the frequencies of the repeated letters.

Here, we have:
- Total letters, \( n = 5 \)
- The letter 'I' appears 2 times.

Thus, the total arrangements are:

\[ \text{Total arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \]

### Step 3: Calculate arrangements where vowels are together
To find the arrangements where the vowels are together, we can treat the vowels (I, I, A) as a single unit or block. This gives us the following arrangement:

- Blocks: (IIA), N, D

Now we have 3 blocks: (IIA), N, D. The arrangements of these blocks can be calculated as:

\[ \text{Arrangements of blocks} = \frac{3!}{2!} = \frac{6}{2} = 3 \]

(We divide by \(2!\) because 'I' is repeated.)

### Step 4: Calculate arrangements of the vowels within the block
The vowels can be arranged within their block (IIA) as follows:

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

### Step 5: Calculate total arrangements with vowels together
Now, the total arrangements where the vowels are together is:

\[ \text{Total with vowels together} = \text{Arrangements of blocks} \times \text{Arrangements of vowels} = 3 \times 3 = 9 \]

### Step 6: Calculate arrangements where vowels are not together
To find the arrangements where the vowels are not together, we subtract the arrangements where the vowels are together from the total arrangements:

\[ \text{Arrangements with vowels not together} = \text{Total arrangements} - \text{Total with vowels together} = 60 - 9 = 51 \]

### Final Answer
Thus, the number of arrangements of the letters in 'INDIA' such that the vowels are never together is **51**.

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