If the letters of the word `MOTHER` are written in all possible orders and these words are written out as in a dictionary, find rank of the word `MOTHER`.

Options

A. `307`
B. `261`
C. `308`
D. `309`

Explanation

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

### Step 1: Arrange the letters in alphabetical order
The letters of the word "MOTHER" are: M, O, T, H, E, R.
Arranging them in alphabetical order gives us: E, H, M, O, R, T.

### Step 2: Count words starting with letters before 'M'
We will count how many words can be formed starting with each letter that comes before 'M' in the alphabetical order.

1. **Words starting with 'E':**
- Remaining letters: H, M, O, R, T (5 letters)
- Number of arrangements = 5! = 120

2. **Words starting with 'H':**
- Remaining letters: E, M, O, R, T (5 letters)
- Number of arrangements = 5! = 120

### Step 3: Count words starting with 'M'
Now we will consider words starting with 'M'.

1. **Fix 'M' and consider the next letter:**
- Next letters in alphabetical order after 'M': E, H, O, R, T.

2. **Words starting with 'M' followed by 'E':**
- Remaining letters: H, O, R, T (4 letters)
- Number of arrangements = 4! = 24

3. **Words starting with 'M' followed by 'H':**
- Remaining letters: E, O, R, T (4 letters)
- Number of arrangements = 4! = 24

4. **Words starting with 'M' followed by 'O':**
- Remaining letters: E, H, R, T (4 letters)
- Number of arrangements = 4! = 24

### Step 4: Fix 'M' and 'O', consider the next letter
Now we will fix 'M' and 'O' and consider the next letters in alphabetical order: E, H, R, T.

1. **Words starting with 'M', 'O', followed by 'E':**
- Remaining letters: H, R, T (3 letters)
- Number of arrangements = 3! = 6

2. **Words starting with 'M', 'O', followed by 'H':**
- Remaining letters: E, R, T (3 letters)
- Number of arrangements = 3! = 6

3. **Words starting with 'M', 'O', followed by 'R':**
- Remaining letters: E, H, T (3 letters)
- Number of arrangements = 3! = 6

### Step 5: Fix 'M', 'O', 'T', consider the last letter
Now we will fix 'M', 'O', 'T' and consider the last letter: E, H, R.

1. **Words starting with 'M', 'O', 'T', followed by 'E':**
- Remaining letters: H, R (2 letters)
- Number of arrangements = 2! = 2

2. **Words starting with 'M', 'O', 'T', followed by 'H':**
- Remaining letters: E, R (2 letters)
- Number of arrangements = 2! = 2

3. **Words starting with 'M', 'O', 'T', followed by 'R':**
- Remaining letters: E, H (2 letters)
- Number of arrangements = 2! = 2

### Step 6: Count the rank of 'MOTHER'
Now we will add all the arrangements counted:

- Words starting with 'E': 120
- Words starting with 'H': 120
- Words starting with 'M' and 'E': 24
- Words starting with 'M' and 'H': 24
- Words starting with 'M' and 'O' and 'E': 6
- Words starting with 'M' and 'O' and 'H': 6
- Words starting with 'M' and 'O' and 'R': 6
- Words starting with 'M', 'O', 'T', 'E': 2
- Words starting with 'M', 'O', 'T', 'H': 2

Now, we add these values:

Total = 120 + 120 + 24 + 24 + 6 + 6 + 6 + 2 + 2 = 310

### Final Rank Calculation
The rank of the word "MOTHER" is 310.