If the letters of the word 'RACHIT' are arranged in all possible ways as listed in dictionary, then the rank of the word 'RACHIT' is
(i) 480
(ii) 481
(iii) 482
(iv) 483

Options

A. 480
B. 481
C. 482
D. 483

Answer

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

### Step 1: Arrange the letters in alphabetical order
The letters of the word 'RACHIT' are A, C, H, I, R, T. Arranging them in alphabetical order gives:
- A, C, H, I, R, T

### Step 2: Count the arrangements starting with letters before 'R'
We need to count how many words can be formed with the letters that come before 'R' in alphabetical order.

1. **Words starting with 'A':**
- Remaining letters: C, H, I, R, T
- Number of arrangements = 5! = 120

2. **Words starting with 'C':**
- Remaining letters: A, H, I, R, T
- Number of arrangements = 5! = 120

3. **Words starting with 'H':**
- Remaining letters: A, C, I, R, T
- Number of arrangements = 5! = 120

4. **Words starting with 'I':**
- Remaining letters: A, C, H, R, T
- Number of arrangements = 5! = 120

### Step 3: Count the arrangements starting with 'R'
Now we consider the arrangements that start with 'R' and find the next letters.

- The next letter after 'R' is 'A', 'C', 'H', 'I', and 'T'.

1. **Words starting with 'RA':**
- Remaining letters: C, H, I, T
- Number of arrangements = 4! = 24

2. **Words starting with 'RC':**
- Remaining letters: A, H, I, T
- Number of arrangements = 4! = 24

3. **Words starting with 'RH':**
- Remaining letters: A, C, I, T
- Number of arrangements = 4! = 24

4. **Words starting with 'RI':**
- Remaining letters: A, C, H, T
- Number of arrangements = 4! = 24

### Step 4: Count the arrangements starting with 'RAC'
Now we consider arrangements starting with 'RAC'.

1. **Words starting with 'RAC':**
- Remaining letters: H, I, T
- Number of arrangements = 3! = 6

### Step 5: Count the arrangements starting with 'RACHI'
Now we consider arrangements starting with 'RACHI'.

1. **Words starting with 'RACHI':**
- Remaining letters: T
- Number of arrangements = 1! = 1

### Step 6: Calculate the total rank
Now, we sum all the arrangements counted:

- Words starting with 'A': 120
- Words starting with 'C': 120
- Words starting with 'H': 120
- Words starting with 'I': 120
- Words starting with 'RA': 24
- Words starting with 'RC': 24
- Words starting with 'RH': 24
- Words starting with 'RI': 24
- Words starting with 'RAC': 6
- Words starting with 'RACHI': 1

Total = 120 + 120 + 120 + 120 + 24 + 24 + 24 + 24 + 6 + 1 = 1, 481

### Final Rank of 'RACHIT'
Thus, the rank of the word 'RACHIT' is 481.

### Answer
The rank of the word 'RACHIT' is **481** (Option ii).

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