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

Options

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

Explanation

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

### Step 1: Count the total number of letters
The word "TROUBLE" consists of 7 letters: T, R, O, U, B, L, E.

### Step 2: Check for repetitions
In the word "TROUBLE," all letters are unique. There are no repeated letters.

### Step 3: Use the formula for permutations
Since all letters are unique, the number of 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!.

Here, n = 7 (the number of letters in "TROUBLE").

### Step 4: Calculate 7!
Now, we need to calculate 7! (7 factorial):

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

### Step 5: Perform the calculation
Calculating step by step:
- 7 × 6 = 42
- 42 × 5 = 210
- 210 × 4 = 840
- 840 × 3 = 2520
- 2520 × 2 = 5040
- 5040 × 1 = 5040

Thus, 7! = 5040.

### Conclusion
The total number of different ways the letters of the word "TROUBLE" can be arranged is **5040**.

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