How many different letter arrangements can be made from thye letters off the word RECOVER ?

Options

A. 1210
B. 5040
C. 1260
D. 1200

Answer

To find the number of different letter arrangements that can be made from the letters of the word "RECOVER", we can follow these steps:

### Step 1: Count the total number of letters
The word "RECOVER" consists of 7 letters: R, E, C, O, V, E, R.

### Step 2: Identify any repeating letters
In the word "RECOVER", the letters R and E are repeated:
- R appears 2 times
- E appears 2 times

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of n items where there are repetitions 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 items,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeating items.

In our case:
- \( n = 7 \) (total letters)
- \( p_1 = 2 \) (for R)
- \( p_2 = 2 \) (for E)

### Step 4: Substitute values into the formula
Substituting the values into the formula gives us:

\[ \text{Number of arrangements} = \frac{7!}{2! \times 2!} \]

### Step 5: Calculate factorials
Now, we calculate the factorials:
- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 6: Substitute the factorial values back into the equation
Now substituting these values back into our equation:

\[ \text{Number of arrangements} = \frac{5040}{2 \times 2} = \frac{5040}{4} = 1260 \]

### Conclusion
Thus, the total number of different letter arrangements that can be made from the letters of the word "RECOVER" is **1260**.

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