IF all the words formed by the word RAINBOW are arranged in a dictionary form then what is the position of the word RAINBOW in that dictionary ?

Options

A. 3136
B. 3631
C. 3361
D. 1363

Answer

To find the position of the word "RAINBOW" when all permutations of its letters are arranged in dictionary order, we can follow these steps:

### Step-by-Step Solution:

1. **List the Letters in Alphabetical Order**:
The letters in "RAINBOW" are R, A, I, N, B, O, W. Arranging them in alphabetical order gives us:
- A, B, I, N, O, R, W

2. **Count the Total Letters**:
The word "RAINBOW" has 7 letters.

3. **Calculate the Number of Words Starting with Each Letter Before 'R'**:
- **Words starting with 'A'**:
- Remaining letters: B, I, N, O, R, W (6 letters)
- Number of permutations = 6! = 720
- **Words starting with 'B'**:
- Remaining letters: A, I, N, O, R, W (6 letters)
- Number of permutations = 6! = 720
- **Words starting with 'I'**:
- Remaining letters: A, B, N, O, R, W (6 letters)
- Number of permutations = 6! = 720
- **Words starting with 'N'**:
- Remaining letters: A, B, I, O, R, W (6 letters)
- Number of permutations = 6! = 720
- **Words starting with 'O'**:
- Remaining letters: A, B, I, N, R, W (6 letters)
- Number of permutations = 6! = 720

4. **Count Words Starting with 'R'**:
Now we consider words starting with 'R'. The next letter must be 'A' (the next letter in "RAINBOW").
- **Words starting with 'RA'**:
- Remaining letters: B, I, N, O, W (5 letters)
- Number of permutations = 5! = 120

5. **Count Words Starting with 'RAB'**:
- **Words starting with 'RAB'**:
- Remaining letters: I, N, O, W (4 letters)
- Number of permutations = 4! = 24

6. **Count Words Starting with 'RAI'**:
- **Words starting with 'RAI'**:
- Remaining letters: B, N, O, W (4 letters)
- Number of permutations = 4! = 24

7. **Count Words Starting with 'RAIN'**:
- **Words starting with 'RAIN'**:
- Remaining letters: B, O, W (3 letters)
- Number of permutations = 3! = 6

8. **Count Words Starting with 'RAINB'**:
- **Words starting with 'RAINB'**:
- Remaining letters: O, W (2 letters)
- Number of permutations = 2! = 2

9. **Count the Position of 'RAINBOW'**:
Now we have counted all the permutations before "RAINBOW":
- From 'A': 720
- From 'B': 720
- From 'I': 720
- From 'N': 720
- From 'O': 720
- From 'RA': 120
- From 'RAB': 24
- From 'RAI': 24
- From 'RAIN': 6
- From 'RAINB': 2

Adding these gives:
\[ 720 + 720 + 720 + 720 + 720 + 120 + 24 + 24 + 6 + 2 = 3606 \]

The word "RAINBOW" itself is the next word, so we add 1:
\[ 3606 + 1 = 3607 \]

### Final Answer:
The position of the word "RAINBOW" in the dictionary is **3607**.