If the letters of the word RACHIT are arranged in all possible ways and these words (with or without meaning) are written as in a dictionary, then find the rank of this word RACHIT.

Options

A. A) 480
B. B) 240
C. C) 481
D. D) 241

Answer

To find the rank of the word "RACHIT" when all its letters are arranged in alphabetical order, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Letters and Their Order**:
The letters in "RACHIT" are A, C, H, I, R, T. Arranging them in alphabetical order gives us:
- A, C, H, I, R, T

2. **Count Words Starting with Letters Before 'R'**:
We need to count how many words can be formed with letters that come before 'R' in the alphabetical order.

- **Fixing 'A'**:
- Remaining letters: C, H, I, R, T (5 letters)
- Number of arrangements = 5! = 120

- **Fixing 'C'**:
- Remaining letters: A, H, I, R, T (5 letters)
- Number of arrangements = 5! = 120

- **Fixing 'H'**:
- Remaining letters: A, C, I, R, T (5 letters)
- Number of arrangements = 5! = 120

- **Fixing 'I'**:
- Remaining letters: A, C, H, R, T (5 letters)
- Number of arrangements = 5! = 120

Adding these counts gives:
\[ 120 + 120 + 120 + 120 = 480 \]

3. **Count Words Starting with 'R'**:
Now we need to consider the arrangements that start with 'R'. The next letter in "RACHIT" is 'A'.

- Fixing 'RA':
- Remaining letters: C, H, I, T (4 letters)
- Number of arrangements = 4! = 24

- Fixing 'RC':
- Remaining letters: A, H, I, T (4 letters)
- Number of arrangements = 4! = 24

- Fixing 'RH':
- Remaining letters: A, C, I, T (4 letters)
- Number of arrangements = 4! = 24

- Fixing 'RI':
- Remaining letters: A, C, H, T (4 letters)
- Number of arrangements = 4! = 24

Adding these counts gives:
\[ 24 + 24 + 24 + 24 = 96 \]

4. **Count the Word 'RACHIT'**:
The next word after all those starting with 'R' and letters before 'A' is 'RACHIT' itself. Therefore, we add 1 for the word 'RACHIT'.

5. **Calculate the Final Rank**:
The total rank of the word "RACHIT" is:
\[ \text{Rank} = 480 + 96 + 1 = 577 \]

### Final Answer:
The rank of the word "RACHIT" is **577**.