To find the rank of the word "MOTHER" when all permutations of its letters are arranged in alphabetical order, we can follow these steps:
### Step 1: Arrange the letters of "MOTHER" in alphabetical order.
The letters of "MOTHER" are: E, H, M, O, R, T.
### Step 2: Count the words that come before "MOTHER".
We will analyze the permutations of the letters that come before "MOTHER" in alphabetical order.
1. **Words starting with E**:
- Remaining letters: H, M, O, R, T
- The number of permutations = 5! = 120.
2. **Words starting with H**:
- Remaining letters: E, M, O, R, T
- The number of permutations = 5! = 120.
3. **Words starting with M**:
- The next letter after M in "MOTHER" is O. We need to consider words starting with M but having letters before O.
- **Words starting with ME**:
- Remaining letters: H, O, R, T
- The number of permutations = 4! = 24.
- **Words starting with MH**:
- Remaining letters: E, O, R, T
- The number of permutations = 4! = 24.
- **Words starting with MO**:
- The next letter after M and O in "MOTHER" is T. We need to consider words starting with MO but having letters before T.
- **Words starting with MOE**:
- Remaining letters: H, R, T
- The number of permutations = 3! = 6.
- **Words starting with MOH**:
- Remaining letters: E, R, T
- The number of permutations = 3! = 6.
- **Words starting with MOR**:
- Remaining letters: E, H, T
- The number of permutations = 3! = 6.
### Step 3: Count the words that match "MOTHER".
Now we can count the rank of "MOTHER" itself:
- The letters of "MOTHER" are M, O, T, H, E, R.
### Step 4: Calculate the total rank.
Now, we sum all the counts:
- Words starting with E: 120
- Words starting with H: 120
- Words starting with ME: 24
- Words starting with MH: 24
- Words starting with MOE: 6
- Words starting with MOH: 6
- Words starting with MOR: 6
- The word "MOTHER": 1
Total rank = 120 + 120 + 24 + 24 + 6 + 6 + 6 + 1 = 307.
### Final Answer:
The rank of the word "MOTHER" is **307**.
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