Threre are 9 red,7 white and 4 black balls in an urn.
In how many different ways can the letters of the word RUMOUR be arranged ?

Options

A. 180
B. 720
C. 30
D. 90

Explanation

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

### Step 1: Count the total number of letters
The word "RUMOUR" consists of 6 letters: R, U, M, O, U, R.

### Step 2: Identify the frequency of each letter
- R appears 2 times
- U appears 2 times
- M appears 1 time
- O 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.

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

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

### Step 6: Substitute the factorial values into the equation
\[ \text{Number of arrangements} = \frac{720}{2 \times 2 \times 1 \times 1} = \frac{720}{4} = 180 \]

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

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