In how many different ways can the letters of the word 'VIRTUAL ' be arranged among themselves ?

Options

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

Explanation

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

### Step 1: Identify the letters in the word
The word "VIRTUAL" consists of the following letters: V, I, R, T, U, A, L.

### Step 2: Count the number of letters
The total number of letters in "VIRTUAL" is 7.

### Step 3: Check for any repeating letters
In the word "VIRTUAL", all the letters are unique. There are no repeating letters.

### Step 4: Use the formula for permutations
Since all the letters are unique, the number of different arrangements (permutations) of the letters can be calculated using the factorial of the number of letters. The formula for the number of arrangements of n distinct objects is given by n!.

### Step 5: Calculate the factorial
For the word "VIRTUAL", we need to calculate 7! (7 factorial):
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

### Step 6: Perform the calculation
Calculating this step by step:
- \( 7 \times 6 = 42 \)
- \( 42 \times 5 = 210 \)
- \( 210 \times 4 = 840 \)
- \( 840 \times 3 = 2520 \)
- \( 2520 \times 2 = 5040 \)
- \( 5040 \times 1 = 5040 \)

Thus, \( 7! = 5040 \).

### Step 7: Conclusion
The total number of different ways the letters of the word "VIRTUAL" can be arranged is **5040**.

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