In how many different ways can the letters of the word 'ASSISTANCE' be arranged?

Options

A. 604800
B. 1814400
C. 1209600
D. None of these

Answer

To find the number of different ways the letters of the word 'ASSISTANCE' can be arranged, we follow these steps:

### Step 1: Count the total number of letters
The word 'ASSISTANCE' has a total of 11 letters.

### Step 2: Identify the frequency of each letter
In the word 'ASSISTANCE', the frequency of each letter is as follows:
- A: 2
- S: 3
- I: 1
- T: 1
- N: 1
- C: 1
- E: 1

### Step 3: Use the formula for permutations of multiset
The formula to calculate the number of arrangements of letters in a word where some letters are repeated 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.

For 'ASSISTANCE', we have:

\[ n = 11, \quad n_A = 2, \quad n_S = 3, \quad n_I = 1, \quad n_T = 1, \quad n_N = 1, \quad n_C = 1, \quad n_E = 1 \]

### Step 4: Substitute values into the formula
Substituting the values into the formula, we get:

\[ \text{Number of arrangements} = \frac{11!}{2! \times 3! \times 1! \times 1! \times 1! \times 1! \times 1!} \]

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

- \( 11! = 39916800 \)
- \( 2! = 2 \)
- \( 3! = 6 \)
- \( 1! = 1 \)

### Step 6: Substitute the factorial values
Now we can substitute these values back into the equation:

\[ \text{Number of arrangements} = \frac{39916800}{2 \times 6 \times 1 \times 1 \times 1 \times 1 \times 1} = \frac{39916800}{12} \]

### Step 7: Perform the division
Now, we perform the division:

\[ \frac{39916800}{12} = 3326400 \]

### Final Answer
Thus, the total number of different ways the letters of the word 'ASSISTANCE' can be arranged is **3326400**.

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