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

Options

A. 297
B. 295
C. 299
D. 301

Answer

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

### Step 1: Arrange the letters in alphabetical order
The letters of the word "MUMBAI" are A, B, I, M, M, U.

### Step 2: Calculate the total arrangements starting with letters before 'M'
1. **Words starting with 'A':**
- Remaining letters: B, I, M, M, U (5 letters)
- Total arrangements = 5! / 2! = 120 / 2 = 60 (since M is repeated)

2. **Words starting with 'B':**
- Remaining letters: A, I, M, M, U (5 letters)
- Total arrangements = 5! / 2! = 120 / 2 = 60

3. **Words starting with 'I':**
- Remaining letters: A, B, M, M, U (5 letters)
- Total arrangements = 5! / 2! = 120 / 2 = 60

Adding these gives:
- Total words before 'M': 60 + 60 + 60 = 180

### Step 3: Fix the first letter as 'M' and calculate for the second letter
Now we consider words starting with 'M'.

1. **Words starting with 'MA':**
- Remaining letters: B, I, M, U (4 letters)
- Total arrangements = 4! = 24

2. **Words starting with 'MB':**
- Remaining letters: A, I, M, U (4 letters)
- Total arrangements = 4! = 24

3. **Words starting with 'MI':**
- Remaining letters: A, B, M, U (4 letters)
- Total arrangements = 4! = 24

4. **Words starting with 'MM':**
- Remaining letters: A, B, I, U (4 letters)
- Total arrangements = 4! = 24

Adding these gives:
- Total words starting with 'M' before 'MUMBAI': 24 + 24 + 24 = 72

### Step 4: Fix the first two letters as 'MU' and calculate for the third letter
Now we consider words starting with 'MU'.

1. **Words starting with 'MUA':**
- Remaining letters: B, I, M (3 letters)
- Total arrangements = 3! = 6

2. **Words starting with 'MUB':**
- Remaining letters: A, I, M (3 letters)
- Total arrangements = 3! = 6

3. **Words starting with 'MUI':**
- Remaining letters: A, B, M (3 letters)
- Total arrangements = 3! = 6

4. **Words starting with 'MUM':**
- Remaining letters: A, B, I (3 letters)
- Total arrangements = 3! = 6

Now we need to fix 'MUM' and calculate for the fourth letter.

### Step 5: Fix the first three letters as 'MUM' and calculate for the fourth letter
Now we consider words starting with 'MUM'.

1. **Words starting with 'MUMA':**
- Remaining letters: B, I (2 letters)
- Total arrangements = 2! = 2

2. **Words starting with 'MUMB':**
- Remaining letters: A, I (2 letters)
- Total arrangements = 2! = 2

3. **Words starting with 'MUMI':**
- Remaining letters: A, B (2 letters)
- Total arrangements = 2! = 2

Finally, we reach 'MUMBAI'.

### Step 6: Calculate the rank
Now we sum all the arrangements calculated:
- Words before 'M': 180
- Words starting with 'MA', 'MB', 'MI', 'MM': 72
- Words starting with 'MUA', 'MUB', 'MUI', 'MUM': 6 + 6 + 6 + 6 = 24
- Words starting with 'MUMA': 2
- Words starting with 'MUMB': 2
- The word 'MUMBAI' itself counts as 1.

Adding these gives:
- Total rank = 180 + 72 + 24 + 2 + 2 + 1 = 281

Thus, the rank of the word 'MUMBAI' is **281**.