Number of ways in which the letters of the word SUCCESSFUL be arranged is

Options

A. `(10!)/(5!)`
B. `(13!)/(5!)`
C. `(10!)/(3!(2!)^(2))`
D. `(10!)/((3!2!)^(2))`

Answer

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

### Step 1: Count the Total Letters
First, we count the total number of letters in the word "SUCCESSFUL".

- The letters are: S, U, C, C, E, S, S, F, U, L
- Total letters = 10

### Step 2: Identify Repeating Letters
Next, we identify the letters that are repeating and their frequencies.

- S appears 3 times
- U appears 2 times
- C appears 2 times
- E appears 1 time
- F appears 1 time
- L appears 1 time

### Step 3: Apply the Formula for Arrangements
The formula to calculate the number of arrangements of letters when there are repetitions is given by:

\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times p_3! \times \ldots} \]

Where:
- \( n \) = total number of letters
- \( p_1, p_2, p_3, \ldots \) = frequencies of the repeating letters

In our case:
- \( n = 10 \)
- \( p_1 = 3 \) (for S)
- \( p_2 = 2 \) (for U)
- \( p_3 = 2 \) (for C)
- \( p_4 = 1 \) (for E)
- \( p_5 = 1 \) (for F)
- \( p_6 = 1 \) (for L)

### Step 4: Substitute Values into the Formula
Now we substitute the values into the formula:

\[ \text{Number of arrangements} = \frac{10!}{3! \times 2! \times 2! \times 1! \times 1! \times 1!} \]

### Step 5: Calculate Factorials
Now we will calculate the factorials:

- \( 10! = 3628800 \)
- \( 3! = 6 \)
- \( 2! = 2 \)
- \( 1! = 1 \)

### Step 6: Substitute and Simplify
Now we substitute these values back into the equation:

\[ \text{Number of arrangements} = \frac{3628800}{6 \times 2 \times 2 \times 1 \times 1 \times 1} \]

Calculating the denominator:

\[ 6 \times 2 \times 2 = 24 \]

So now we have:

\[ \text{Number of arrangements} = \frac{3628800}{24} = 151200 \]

### Final Answer
Therefore, the number of ways in which the letters of the word "SUCCESSFUL" can be arranged is **151200**.

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