In how many ways, the letters of the word 'ARMOUR' can be arranged?

Options

A. 720
B. 300
C. 640
D. None of the above

Answer

To find the number of ways the letters of the word 'ARMOUR' can be arranged, we can follow these steps:

### Step-by-Step Solution:

1. **Count the Total Letters**:
The word 'ARMOUR' consists of 6 letters: A, R, M, O, U, R.

2. **Identify Repeating Letters**:
In the word 'ARMOUR', the letter 'R' is repeated. It appears 2 times.

3. **Use the Formula for Arrangements**:
The formula to calculate the number of arrangements of letters in a word where some letters are repeated is given by:
\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]
where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

4. **Substitute the Values**:
Here, \( n = 6 \) (total letters), and the frequency of 'R' is 2. Thus, we have:
\[ \text{Number of arrangements} = \frac{6!}{2!} \]

5. **Calculate Factorials**:
- Calculate \( 6! \):
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]
- Calculate \( 2! \):
\[ 2! = 2 \times 1 = 2 \]

6. **Final Calculation**:
Now substitute the values back into the formula:
\[ \text{Number of arrangements} = \frac{720}{2} = 360 \]

### Conclusion:
The total number of ways the letters of the word 'ARMOUR' can be arranged is **360**.

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