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

Options

A. a. 234000
B. b. 123000
C. c. 232300
D. d. 302400

Answer

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

### Step 1: Count the total number of letters
The word "RECUPERATE" consists of 10 letters.

### Step 2: Identify the frequency of each letter
- R appears 2 times
- E appears 3 times
- C appears 1 time
- U appears 1 time
- P appears 1 time
- T appears 1 time
- A appears 1 time

### Step 3: Use the formula for permutations of multiset
The formula to find the number of distinct arrangements of letters when there are repetitions is given by:

\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times n_3! \times \ldots} \]

Where:
- \(n\) is the total number of letters,
- \(n_1, n_2, n_3, \ldots\) are the frequencies of the repeated letters.

### Step 4: Substitute the values into the formula
Here, \(n = 10\) (total letters), \(n_1 = 2\) (for R), and \(n_2 = 3\) (for E). The other letters appear once, so their factorials will be 1 and will not affect the calculation.

Thus, the formula becomes:

\[ \text{Number of arrangements} = \frac{10!}{2! \times 3!} \]

### Step 5: Calculate the factorials
- \(10! = 3628800\)
- \(2! = 2\)
- \(3! = 6\)

### Step 6: Perform the calculation
Now, substituting the values:

\[ \text{Number of arrangements} = \frac{3628800}{2 \times 6} = \frac{3628800}{12} = 302400 \]

### Conclusion
The total number of distinct arrangements of the letters in the word "RECUPERATE" is **302400**.

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