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

Options

A. 840
B. 1680
C. 2520
D. 5040

Explanation

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

### Step 1: Count the total number of letters
The word "BELIEVE" consists of 7 letters in total.

### Step 2: Identify any repeating letters
In the word "BELIEVE," the letter 'E' appears 3 times. The other letters (B, L, I, V) appear only once.

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

\[ \text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]

Where:
- \( n \) is the total number of letters,
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the repeating letters.

For "BELIEVE":
- \( n = 7 \) (total letters),
- The letter 'E' repeats 3 times, so \( n_E = 3 \).

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

\[ \text{Number of arrangements} = \frac{7!}{3!} \]

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

- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

### Step 6: Compute the number of arrangements
Now we can compute the number of arrangements:

\[ \text{Number of arrangements} = \frac{5040}{6} = 840 \]

### Final Answer
Thus, the number of different ways to arrange the letters of the word "BELIEVE" is **840**.

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