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 with the letters of the word "INDIA", we can follow these steps:

### Step 1: Identify the letters and their frequencies
The word "INDIA" consists of the following letters:
- I: 2 times
- N: 1 time
- D: 1 time
- A: 1 time

### Step 2: Calculate the total number of letters
The total number of letters in "INDIA" is 5.

### Step 3: Use the formula for permutations of multiset
The formula for 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 items (letters in this case),
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the distinct items.

In our case:
- \( n = 5 \) (total letters)
- The frequencies are:
- For I: \( n_1 = 2 \)
- For N: \( n_2 = 1 \)
- For D: \( n_3 = 1 \)
- For A: \( n_4 = 1 \)

### Step 4: Substitute the values into the formula
Now, substituting the values into the formula gives us:

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

### Step 5: Calculate the factorials
Now we calculate the factorials:
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
- \( 2! = 2 \times 1 = 2 \)
- \( 1! = 1 \)

### Step 6: Substitute the factorial values
Now substitute these values back into the equation:

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

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

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