In how many different ways can the letters of the word 'KURUKSHETRA' be arranged?

Options

A. `4497600`
B. `4979600`
C. `4989600`
D. `4789600`

Answer

To find the number of different ways the letters of the word 'KURUKSHETRA' can be arranged, we need to follow these steps:

### Step 1: Count the total number of letters
The word 'KURUKSHETRA' consists of 11 letters.

### Step 2: Identify the repeated letters
In the word 'KURUKSHETRA', the letters are as follows:
- K: 2 times
- U: 2 times
- R: 2 times
- H: 1 time
- S: 1 time
- E: 1 time
- T: 1 time
- A: 1 time

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of letters in a word where some letters are repeated is given by:

\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times p_3! \times \ldots} \]

Where:
- \( n \) is the total number of letters,
- \( p_1, p_2, p_3, \ldots \) are the frequencies of the repeated letters.

In our case:
- \( n = 11 \) (total letters)
- \( p_1 = 2 \) (for K)
- \( p_2 = 2 \) (for U)
- \( p_3 = 2 \) (for R)

### Step 4: Substitute the values into the formula
Now we can substitute the values into the formula:

\[ \text{Number of arrangements} = \frac{11!}{2! \times 2! \times 2!} \]

### Step 5: Calculate the factorials
Now we calculate the factorials:

- \( 11! = 39916800 \)
- \( 2! = 2 \)

Thus,

\[ \text{Number of arrangements} = \frac{39916800}{2 \times 2 \times 2} = \frac{39916800}{8} = 4989600 \]

### Final Answer
The number of different ways the letters of the word 'KURUKSHETRA' can be arranged is **4,989,600**.

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