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

Options

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

Answer

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

### Step 1: Identify the total number of letters
The word "PRAYER" consists of 6 letters: P, R, A, Y, E, R.

### Step 2: Count the frequency of each letter
In the word "PRAYER":
- P appears 1 time
- R appears 2 times
- A appears 1 time
- Y appears 1 time
- E appears 1 time

### Step 3: Use the formula for permutations of multiset
The formula for finding 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,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

### Step 4: Apply the formula
Here, \( n = 6 \) (total letters), and the frequencies are:
- R appears 2 times, so \( p_1 = 2! \)
- All other letters appear 1 time, so their factorials will be \( 1! \) (which equals 1 and does not affect the calculation).

Thus, we can write:

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

### Step 5: Calculate \( 6! \) and \( 2! \)
- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 6: Substitute and simplify
Now substituting the values into the formula:

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

### Final Answer
The total number of different ways the letters of the word "PRAYER" can be arranged is **360**.

---