How many different strings of letters can be made by recording the letters of the word SUCCESS ?

Options

A. 20
B. 30
C. 40
D. 420

Answer

To find the number of different strings of letters that can be made by rearranging the letters of the word "SUCCESS", we will follow these steps:

### Step 1: Identify the total number of letters
The word "SUCCESS" consists of 7 letters.

### Step 2: Identify the frequency of each letter
In the word "SUCCESS", the frequency of each letter is as follows:
- S: 3 times
- U: 1 time
- C: 2 times
- E: 1 time

### Step 3: Use the formula for arrangements of letters
The formula for the number of arrangements of n objects where some objects are identical is given by:

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

where:
- \( n \) is the total number of letters,
- \( n_1, n_2, \ldots, n_m \) are the frequencies of the identical letters.

### Step 4: Substitute the values into the formula
Here, we have:
- \( n = 7 \) (total letters)
- \( n_1 = 3 \) (for S)
- \( n_2 = 2 \) (for C)
- \( n_3 = 1 \) (for U)
- \( n_4 = 1 \) (for E)

So, the number of arrangements becomes:

\[ \text{Number of arrangements} = \frac{7!}{3! \times 2! \times 1! \times 1!} \]

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

### Step 6: Substitute the factorial values back into the formula
Now substituting these values into the formula gives:

\[ \text{Number of arrangements} = \frac{5040}{6 \times 2 \times 1 \times 1} = \frac{5040}{12} \]

### Step 7: Perform the final calculation
Calculating the final value:

\[ \frac{5040}{12} = 420 \]

### Conclusion
Therefore, the number of different strings of letters that can be made by rearranging the letters of the word "SUCCESS" is **420**.

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