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

Options

A. 5040
B. 2040
C. 2510
D. None of the above

Answer

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

### Step 1: Identify the total number of letters
The word "FINANCE" consists of 7 letters: F, I, N, A, N, C, E.

### Step 2: Identify any repeating letters
In the word "FINANCE", the letter 'N' appears twice.

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of letters in a word where some letters may repeat is given by:

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

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

### Step 4: Calculate the factorials
- Here, \( n = 7 \) (total letters),
- The letter 'N' repeats 2 times.

Thus, we can write:

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

### Step 5: Calculate \( 7! \) and \( 2! \)
- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 6: Substitute the values into the formula
Now substituting the values we calculated:

\[ \text{Number of arrangements} = \frac{5040}{2} = 2520 \]

### Final Answer
The total number of different ways to arrange the letters of the word "FINANCE" is **2520**.

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