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

Options

A. 840
B. 5040
C. 1260
D. 2520

Explanation

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

### Step 1: Identify the letters and their frequency
The word "VENTURE" consists of 7 letters: V, E, N, T, U, R, E. Here, we notice that the letter 'E' appears twice while all other letters appear once.

### Step 2: Use the formula for permutations of multiset
The formula for the number of arrangements of letters in a word where some letters are repeated is given by:

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

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

### Step 3: Plug in the values
In our case:
- Total letters \( n = 7 \) (V, E, N, T, U, R, E)
- The letter 'E' appears 2 times, so \( p_1 = 2 \) (for E)
- All other letters appear once, but since they are unique, their factorials will be 1 and won't affect the calculation.

Thus, we have:

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

### Step 4: Calculate the factorials
Now, we 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 5: Compute the final result
Now, substitute these values back into the formula:

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

### Final Answer
Thus, the letters of the word "VENTURE" can be arranged in **2520 different ways**.

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