No of words formed with the letters of the word 'INDIA' is:

Options

A. 120
B. 60
C. 40
D. None of these

Answer

To find the number of distinct words that can be formed using the letters of the word "INDIA", we will 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 to find the number of distinct arrangements of letters when there are repetitions is given by:

\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

where:
- \( n \) is the total number of letters,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

### Step 4: Substitute the values into the formula
In our case:
- \( n = 5 \) (total letters)
- The letter 'I' repeats 2 times, while 'N', 'D', and 'A' each appear once.

Thus, we have:

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

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

### Step 6: Substitute the factorial values
Now substituting the values into the equation:

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

### Conclusion
The total number of distinct words that can be formed with the letters of the word "INDIA" is **60**.

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