The letters of the word "KANPUR" are arranged in all possible ways as in a dictionary, the rank of the word 'KANPUR' from last is

Options

A. 121
B. 122
C. 598
D. 599

Answer

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

### Step 1: Identify the total number of letters
The word "KANPUR" has 6 distinct letters: K, A, N, P, U, R.

### Step 2: Calculate the total arrangements
Since all letters are distinct, the total number of arrangements of the letters is given by \(6!\) (factorial of 6).
\[ 6! = 720 \]

### Step 3: Arrange the letters in alphabetical order
The alphabetical order of the letters in "KANPUR" is:
A, K, N, P, R, U

### Step 4: Count arrangements starting with letters before 'K'
1. **Words starting with 'A':**
- Remaining letters: K, N, P, R, U (5 letters)
- Arrangements = \(5! = 120\)

So, there are 120 words starting with 'A'.

### Step 5: Count arrangements starting with 'K'
Now we focus on words starting with 'K'. The next letter in "KANPUR" is 'A'.

1. **Words starting with 'KA':**
- Remaining letters: N, P, R, U (4 letters)
- Arrangements = \(4! = 24\)

So, there are 24 words starting with 'KA'.

### Step 6: Count arrangements starting with 'KN'
Next, we consider words starting with 'KN'.

1. **Words starting with 'KN':**
- Remaining letters: A, P, R, U (4 letters)
- Arrangements = \(4! = 24\)

So, there are 24 words starting with 'KN'.

### Step 7: Count arrangements starting with 'KNP'
Next, we consider words starting with 'KNP'.

1. **Words starting with 'KNP':**
- Remaining letters: A, R, U (3 letters)
- Arrangements = \(3! = 6\)

So, there are 6 words starting with 'KNP'.

### Step 8: Count arrangements starting with 'KNA'
Next, we consider words starting with 'KNA'.

1. **Words starting with 'KNA':**
- Remaining letters: P, R, U (3 letters)
- Arrangements = \(3! = 6\)

So, there are 6 words starting with 'KNA'.

### Step 9: Count arrangements starting with 'KNPA'
Next, we consider words starting with 'KNPA'.

1. **Words starting with 'KNPA':**
- Remaining letters: R, U (2 letters)
- Arrangements = \(2! = 2\)

So, there are 2 words starting with 'KNPA'.

### Step 10: Count arrangements starting with 'KNPR'
Finally, we consider words starting with 'KNPR'.

1. **Words starting with 'KNPR':**
- Remaining letters: U (1 letter)
- Arrangements = \(1! = 1\)

So, there is 1 word starting with 'KNPR'.

### Step 11: Add up all the arrangements
Now, we can calculate the rank of "KANPUR":
- Words starting with 'A': 120
- Words starting with 'KA': 24
- Words starting with 'KN': 24
- Words starting with 'KNP': 6
- Words starting with 'KNA': 6
- Words starting with 'KNPA': 2
- Words starting with 'KNPR': 1

Adding these gives:
\[ 120 + 24 + 24 + 6 + 6 + 2 + 1 = 183 \]

### Step 12: Calculate the rank of "KANPUR"
The rank of "KANPUR" from the beginning is \(183 + 1 = 184\).

### Step 13: Calculate the rank from the last
To find the rank from the last, we subtract the rank from the total arrangements:
\[ \text{Rank from last} = 720 - 184 + 1 = 537 \]

### Final Answer
The rank of the word "KANPUR" from the last is **537**.