If all letters of word ‘MOTHER’ are written in all possible orders and the word so formed are arranged in a dictionary order, then find the rank of word ‘MOTHER’?

Answer

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

### Step-by-Step Solution:

1. **Identify the letters in alphabetical order**:
The letters in "MOTHER" are: E, H, M, O, R, T.
In alphabetical order, they are: E, H, M, O, R, T.

2. **Count words starting with letters before 'M'**:
- **Words starting with 'E'**: The remaining letters are H, M, O, R, T. The number of arrangements is:
\[ 5! = 120 \]
- **Words starting with 'H'**: The remaining letters are E, M, O, R, T. The number of arrangements is:
\[ 5! = 120 \]

Total for words starting with 'E' and 'H':
\[ 120 + 120 = 240 \]

3. **Count words starting with 'M'**:
Now we consider words starting with 'M'. The next letter in "MOTHER" is 'O'.

- **Words starting with 'ME'**: The remaining letters are H, O, R, T. The number of arrangements is:
\[ 4! = 24 \]

- **Words starting with 'MH'**: The remaining letters are E, O, R, T. The number of arrangements is:
\[ 4! = 24 \]

Total for words starting with 'ME' and 'MH':
\[ 240 + 24 + 24 = 288 \]

4. **Count words starting with 'MO'**:
Now we consider words starting with 'MO'. The next letter in "MOTHER" is 'T'.

- **Words starting with 'MOE'**: The remaining letters are H, R, T. The number of arrangements is:
\[ 3! = 6 \]

- **Words starting with 'MOH'**: The remaining letters are E, R, T. The number of arrangements is:
\[ 3! = 6 \]

- **Words starting with 'MOT'**: The remaining letters are H, E, R. The number of arrangements is:
\[ 3! = 6 \]

Total for words starting with 'MOE', 'MOH', and 'MOT':
\[ 288 + 6 + 6 + 6 = 306 \]

5. **Count words starting with 'MOT'**:
Now we consider words starting with 'MOT'. The next letter in "MOTHER" is 'H'.

- **Words starting with 'MOTE'**: The remaining letters are H, R. The number of arrangements is:
\[ 2! = 2 \]

Now we have reached 'MOTH', and the next letter is 'E', which is the first arrangement of 'MOTHER'.

6. **Final Calculation**:
The total number of words before "MOTHER" is:
\[ 306 + 2 = 308 \]
Therefore, the rank of the word "MOTHER" is:
\[ 308 + 1 = 309 \]

### Conclusion:
The rank of the word "MOTHER" in the dictionary order is **309**.