In hwo many different ways can the letters of the word 'RUMOUR' be arranged?

Options

A. 180
B. 90
C. 30
D. None of these

Answer

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

### Step 1: Identify the letters and their frequencies
The word "RUMOUR" consists of the following letters:
- R: 2 times
- U: 2 times
- M: 1 time
- O: 1 time

### Step 2: Calculate the total number of letters
The total number of letters in "RUMOUR" is 6.

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of n items where there are groups of indistinguishable items 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 items,
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the indistinguishable items.

In our case:
- \( n = 6 \) (total letters)
- \( n_1 = 2 \) (for R)
- \( n_2 = 2 \) (for U)
- \( n_3 = 1 \) (for M)
- \( n_4 = 1 \) (for O)

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

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

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

### Step 6: Substitute the factorials into the equation
Substituting these values back into the equation gives us:

\[ \text{Number of arrangements} = \frac{720}{2 \times 2 \times 1 \times 1} = \frac{720}{4} = 180 \]

### Final Answer
Thus, the letters of the word "RUMOUR" can be arranged in **180 different ways**.

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