Find the number of words formed with the letters of the word 'INDIA'.

Answer

To find the number of distinct words that can be formed using the letters of the word "INDIA", we can follow these steps:

### Step 1: Identify the total number of letters
The word "INDIA" consists of 5 letters: I, N, D, I, A.

### Step 2: Count the frequency of each letter
In the word "INDIA":
- The letter I appears 2 times.
- The letters N, D, and A each appear 1 time.

### Step 3: Use the formula for permutations of multiset
The formula to calculate the number of distinct permutations of a multiset is given by:

\[ \text{Number of permutations} = \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]

where:
- \( n \) is the total number of letters,
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the distinct letters.

### Step 4: Apply the formula
Here, we have:
- Total letters \( n = 5 \) (I, N, D, I, A)
- Frequencies: I appears 2 times, N appears 1 time, D appears 1 time, A appears 1 time.

Using the formula:

\[ \text{Number of permutations} = \frac{5!}{2! \times 1! \times 1! \times 1!} \]

### Step 5: Calculate the factorials
Calculating the factorials:
- \( 5! = 120 \)
- \( 2! = 2 \)
- \( 1! = 1 \) (for N, D, and A)

### Step 6: Substitute the values into the formula
Now substituting the values we calculated:

\[ \text{Number of permutations} = \frac{120}{2 \times 1 \times 1 \times 1} = \frac{120}{2} = 60 \]

### Final Answer
Thus, the total number of distinct words that can be formed with the letters of the word "INDIA" is **60**.

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