How many arrangements can be made of the letters of the word 'ARRANGEMENT'?

Options

A. 2492800
B. 249300
C. `2494800`
D. None of these

Answer

To find the number of arrangements of the letters in the word "ARRANGEMENT", we will follow these steps:

### Step 1: Count the total number of letters
The word "ARRANGEMENT" consists of 11 letters.

### Step 2: Identify the repeating letters
In the word "ARRANGEMENT", the letters and their frequencies are as follows:
- A: 2
- R: 2
- N: 2
- G: 1
- E: 2
- M: 1
- T: 1

### Step 3: Apply the formula for permutations of multiset
The formula for finding the number of arrangements of letters in a word where some letters are repeated is given by:

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

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

### Step 4: Substitute the values into the formula
Here, \( n = 11 \) (total letters) and the frequencies of the repeating letters are:
- A: 2
- R: 2
- N: 2
- E: 2

Thus, we can write:

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

### Step 5: Calculate the factorials
Now we need to calculate \( 11! \) and \( 2! \):

- \( 11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39916800 \)
- \( 2! = 2 \)

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

\[ \text{Number of arrangements} = \frac{39916800}{2 \times 2 \times 2 \times 2} = \frac{39916800}{16} \]

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

\[ \frac{39916800}{16} = 2494800 \]

### Conclusion
Thus, the total number of arrangements of the letters in the word "ARRANGEMENT" is **2494800**.

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