In how many ways can the letters of the word REPEAT be arranged ?

Options

A. a. 2240
B. b. 232230
C. c. 360
D. d. 235460

Answer

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

### Step 1: Count the total number of letters
The word "REPEAT" consists of 6 letters: R, E, P, E, A, T.

### Step 2: Identify the frequency of each letter
In the word "REPEAT":
- R appears 1 time
- E appears 2 times
- P appears 1 time
- A appears 1 time
- T appears 1 time

### Step 3: Use the formula for permutations of multiset
The formula to calculate the number of arrangements of letters when there are repeating 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 repeating letters.

### Step 4: Apply the formula
Here, \( n = 6 \) (total letters), and the frequencies are:
- E appears 2 times, so \( p_1 = 2! \)
- R, P, A, and T each appear 1 time, so their factorials are \( 1! \) (which equals 1 and does not affect the product).

Thus, we can write:

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

### Step 5: Calculate the factorials
Now we calculate \( 6! \) and \( 2! \):

- \( 6! = 720 \)
- \( 2! = 2 \)

### Step 6: Perform the division
Now we divide \( 720 \) by \( 2 \):

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

### Final Answer
The total number of ways to arrange the letters of the word "REPEAT" is **360**.

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