In how many ways can the letters of the word RUSSIA be arranged?

Options

A. 360
B. 60
C. 180
D. 64

Answer

To find the number of ways the letters of the word "RUSSIA" can be arranged, we will follow these steps:

### Step 1: Identify the total number of letters
The word "RUSSIA" consists of 6 letters: R, U, S, S, I, A.

### Step 2: Count the frequency of each letter
In the word "RUSSIA":
- R appears 1 time
- U appears 1 time
- S appears 2 times
- I appears 1 time
- A appears 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:

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

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

### Step 4: Apply the formula
Here, \( n = 6 \) (total letters), and the frequencies are:
- For R: \( n_1 = 1! \)
- For U: \( n_2 = 1! \)
- For S: \( n_3 = 2! \)
- For I: \( n_4 = 1! \)
- For A: \( n_5 = 1! \)

Thus, we can substitute into the formula:

\[ \text{Number of arrangements} = \frac{6!}{1! \times 1! \times 2! \times 1! \times 1!} \]

### Step 5: Calculate the factorials
Now we calculate the factorials:
- \( 6! = 720 \)
- \( 1! = 1 \)
- \( 2! = 2 \)

### Step 6: Substitute the values
Now substituting these values into the formula:

\[ \text{Number of arrangements} = \frac{720}{1 \times 1 \times 2 \times 1 \times 1} = \frac{720}{2} = 360 \]

### Conclusion
Therefore, the total number of ways to arrange the letters of the word "RUSSIA" is **360**.

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