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

Options

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

Answer

To find the number of different ways the letters of the word "BELIEVE" can be arranged, we will follow these steps:

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

### Step 2: Identify the 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 arrangements of letters
The formula to calculate the arrangements of letters when there are repeating letters 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: Substitute the values into the formula
Here, \( n = 7 \) (total letters), and since 'E' repeats 3 times, we have:

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

### Step 5: Calculate the factorials
Now we calculate \( 7! \) and \( 3! \):

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

### Step 6: Divide the factorials
Now we can substitute these values back into the formula:

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

### Conclusion
Thus, the number of different ways the letters of the word "BELIEVE" can be arranged is **840**.

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