A dictionary compriing of four lettered words is formed by arranging individually the letters of the words SLOW and FAST. What is rank of the word SLOW?

Answer

To find the rank of the word "SLOW" when all the permutations of the letters from the words "SLOW" and "FAST" are arranged in alphabetical order, we can follow these steps:

### Step 1: List the letters in alphabetical order
The letters of the words "SLOW" and "FAST" are S, L, O, W, F, A, T.
Arranging these letters alphabetically gives us:
A, F, L, O, S, T, W

### Step 2: Count the permutations before "SLOW"
We will count how many permutations come before the word "SLOW".

1. **Fix the first letter as 'A'**:
- Remaining letters: F, L, O, S, T, W
- Number of permutations = 6! = 720

2. **Fix the first letter as 'F'**:
- Remaining letters: A, L, O, S, T, W
- Number of permutations = 6! = 720

3. **Fix the first letter as 'L'**:
- Remaining letters: A, F, O, S, T, W
- Number of permutations = 6! = 720

4. **Fix the first letter as 'O'**:
- Remaining letters: A, F, L, S, T, W
- Number of permutations = 6! = 720

5. **Fix the first letter as 'S'**:
- Now we need to look at the second letter. The second letter in "SLOW" is 'L'.
- Before 'L', we can have:
- **Fix the second letter as 'A'**:
- Remaining letters: F, L, O, T, W
- Number of permutations = 5! = 120
- **Fix the second letter as 'F'**:
- Remaining letters: A, L, O, T, W
- Number of permutations = 5! = 120

6. **Fix the second letter as 'L'**:
- Now we need to look at the third letter. The third letter in "SLOW" is 'O'.
- Before 'O', we can have:
- **Fix the third letter as 'A'**:
- Remaining letters: F, L, T, W
- Number of permutations = 4! = 24
- **Fix the third letter as 'F'**:
- Remaining letters: A, L, T, W
- Number of permutations = 4! = 24

7. **Fix the third letter as 'O'**:
- Now we need to look at the fourth letter. The fourth letter in "SLOW" is 'W'.
- Before 'W', we can have:
- **Fix the fourth letter as 'A'**:
- Remaining letters: F, L, T
- Number of permutations = 3! = 6
- **Fix the fourth letter as 'F'**:
- Remaining letters: A, L, T
- Number of permutations = 3! = 6
- **Fix the fourth letter as 'L'**:
- Remaining letters: A, F, T
- Number of permutations = 3! = 6

### Step 3: Calculate the total permutations before "SLOW"
Now we sum all the permutations counted:
- For A: 720
- For F: 720
- For L: 720
- For O: 720
- For S with A: 120
- For S with F: 120
- For SL with A: 24
- For SL with F: 24
- For SLO with A: 6
- For SLO with F: 6
- For SLL with A: 6

Total = 720 + 720 + 720 + 720 + 120 + 120 + 24 + 24 + 6 + 6 + 6 = 2,586

### Step 4: Add 1 for the rank of "SLOW"
Since we want the rank of "SLOW", we add 1 to the total permutations before it:
Rank of SLOW = 2586 + 1 = 2587

### Final Answer
The rank of the word "SLOW" is **2587**.

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