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

Options

A. a. 181800
B. b. 818100
C. c. 108108
D. d. 10810800

Answer

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

### Step 1: Count the total number of letters
The word "ASSASSINATION" consists of 13 letters.

### Step 2: Identify the frequency of each letter
We need to count how many times each letter appears in the word:
- A: 3 times
- S: 4 times
- I: 2 times
- N: 2 times
- T: 1 time
- O: 1 time

### Step 3: Use the formula for permutations of multiset
The formula to find the number of 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.

In our case:
- \( n = 13 \)
- The frequencies are:
- A: 3
- S: 4
- I: 2
- N: 2
- T: 1
- O: 1

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

\[ \text{Number of arrangements} = \frac{13!}{3! \times 4! \times 2! \times 2! \times 1! \times 1!} \]

### Step 5: Calculate the factorials
Now we calculate the factorials:
- \( 13! = 6227020800 \)
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 2! = 2 \)
- \( 1! = 1 \)

### Step 6: Calculate the denominator
Now calculate the denominator:

\[ 3! \times 4! \times 2! \times 2! \times 1! \times 1! = 6 \times 24 \times 2 \times 2 \times 1 \times 1 = 576 \]

### Step 7: Divide to find the number of arrangements
Now we divide the total arrangements by the product of the factorials:

\[ \text{Number of arrangements} = \frac{6227020800}{576} = 10810800 \]

### Final Answer
The number of ways to arrange the letters of the word "ASSASSINATION" is **10,810,800**.

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