how many ways the letters of the word Rainbow be arranged ?

Options

A. 5040
B. 4050
C. 3040
D. 8040

Answer

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

### Step 1: Identify the number of letters in the word "Rainbow".
The word "Rainbow" consists of 7 distinct letters: R, A, I, N, B, O, W.

### Step 2: Use the formula for permutations of distinct objects.
The formula for the number of arrangements (permutations) of n distinct objects is given by n!.

### Step 3: Calculate the factorial of the number of letters.
Since there are 7 letters in "Rainbow", we need to calculate 7! (7 factorial).

\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

### Step 4: Perform the multiplication.
Calculating step by step:

- 7 × 6 = 42
- 42 × 5 = 210
- 210 × 4 = 840
- 840 × 3 = 2520
- 2520 × 2 = 5040
- 5040 × 1 = 5040

### Step 5: Conclusion
The total number of ways to arrange the letters of the word "Rainbow" is 5040.

### Final Answer:
There are 5040 ways to arrange the letters of the word "Rainbow".

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