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

Options

A. 120
B. 720
C. 360
D. 540

Answer

To find the number of different ways to arrange the letters of the word "DESIRE", we can follow these steps:

### Step 1: Identify the total number of letters
The word "DESIRE" has 6 letters: D, E, S, I, R, E.

### Step 2: Count the frequency of each letter
In the word "DESIRE":
- D appears 1 time
- E appears 2 times
- S appears 1 time
- I appears 1 time
- R appears 1 time

### Step 3: Use the formula for permutations of multiset
The formula to find the number of arrangements of letters when there are repeated letters 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,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

For "DESIRE":
- Total letters \( n = 6 \)
- The letter E is repeated 2 times.

Thus, the formula becomes:

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

### Step 4: Calculate \( 6! \) and \( 2! \)
Now we calculate \( 6! \) and \( 2! \):

\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]
\[ 2! = 2 \times 1 = 2 \]

### Step 5: Substitute the values into the formula
Now we substitute these values into our formula:

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

### Conclusion
The total number of different ways to arrange the letters of the word "DESIRE" is **360**.