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 arranged in alphabetical order, we will follow these steps:

### Step-by-Step Solution:

1. **List the Letters in Alphabetical Order**:
The letters of the word "MUMBAI" are A, B, I, M, M, U.

2. **Count the Total Arrangements Starting with Letters Before 'M'**:
- **Starting with 'A'**: The remaining letters are B, I, M, M, U.
- Total arrangements = \( \frac{5!}{2!} = \frac{120}{2} = 60 \)
- **Starting with 'B'**: The remaining letters are A, I, M, M, U.
- Total arrangements = \( \frac{5!}{2!} = \frac{120}{2} = 60 \)
- **Starting with 'I'**: The remaining letters are A, B, M, M, U.
- Total arrangements = \( \frac{5!}{2!} = \frac{120}{2} = 60 \)

**Total arrangements before 'M'**: \( 60 + 60 + 60 = 180 \)

3. **Consider Arrangements Starting with 'M'**:
Now we consider arrangements starting with 'M'. The next letter can be A, B, I, M, or U.

- **Starting with 'MA'**: Remaining letters are B, I, M, U.
- Total arrangements = \( 4! = 24 \)
- **Starting with 'MB'**: Remaining letters are A, I, M, U.
- Total arrangements = \( 4! = 24 \)
- **Starting with 'MI'**: Remaining letters are A, B, M, U.
- Total arrangements = \( 4! = 24 \)

**Total arrangements starting with 'M' and letters before 'MU'**:
\( 24 + 24 + 24 = 72 \)

4. **Now Consider 'MU'**:
The next letter can be A, B, or M.

- **Starting with 'MUA'**: Remaining letters are B, I, M.
- Total arrangements = \( 3! = 6 \)
- **Starting with 'MUB'**: Remaining letters are A, I, M.
- Total arrangements = \( 3! = 6 \)
- **Starting with 'MUM'**: Remaining letters are A, B, I.
- Now we need to find the arrangements that lead to 'MUMBAI'.

5. **Arrangements Starting with 'MUM'**:
The next letter can be A, B, or I.

- **Starting with 'MUMA'**: Remaining letters are B, I.
- Total arrangements = \( 2! = 2 \)
- **Starting with 'MUMB'**: Remaining letters are A, I.
- Total arrangements = \( 2! = 2 \)
- **Starting with 'MUMI'**: The next letter can only be A or B.
- We are now at 'MUMBA' which is the prefix of 'MUMBAI'.

6. **Arrangements for 'MUMBA'**:
- The next letter is 'I', and we have reached 'MUMBAI'.

7. **Calculate the Total Rank**:
Adding all the arrangements we calculated:
- Before 'M': 180
- Starting with 'M' before 'MU': 72
- Starting with 'MU' before 'MUM': 12 (6 + 6)
- Finally, 'MUMBA' itself counts as 1.

**Total Rank** = \( 180 + 72 + 12 + 1 = 265 \)

### Final Rank:
The rank of the word "MUMBAI" is **265**.