rank of edwise in permutation and combination

The rank of a word in permutations refers to its position when all permutations of its characters are listed in lexicographic (alphabetical) order. To find the rank of the word "EDWISE," we'll follow these steps:

1. **Arrange the Letters in Alphabetical Order:**  
   First, sort the letters of "EDWISE" alphabetically:
   - The sorted order is: **E, D, I, S, W, E**

2. **Fix Each Character and Calculate the Number of Words Possible:**

   Let's find out how many words start with each character that comes before "EDWISE" in lexicographical order:

   - **First Letter - E:**
     - Fix "D" (since "D" comes before "E"):
       - Remaining letters: E, I, S, W, E
       - Number of permutations of the remaining letters = \(\frac{5!}{2!} = \frac{120}{2} = 60\)

     So, there are 60 permutations that start with "D."

   - Now, we fix "E" and move to the next letter:
     - **Second Letter - E (fixed):**
       - Fix "E" (first occurrence of E):
         - Remaining letters: D, I, S, W, E

     - **Third Letter - E (fixed):**
       - Fix "D" (since "D" comes before "E"):
         - Remaining letters: E, I, S, W
         - Number of permutations = 4! = 24

       So, there are 24 permutations that start with "ED."

   - Now, we fix "ED" and move to the next letter:
     - **Fourth Letter - I:**
       - The remaining letters are E, S, W
       - Number of permutations = 3! = 6

     So, there are 6 permutations that start with "EDI."

   - Now, we fix "EDI" and move to the next letter:
     - **Fifth Letter - S:**
       - The remaining letters are E, W
       - Number of permutations = 2! = 2

     So, there are 2 permutations that start with "EDIS."

   - Finally, we fix "EDIS":
     - **Sixth Letter - W:**
       - The remaining letter is E
       - Number of permutations = 1! = 1

     So, there is 1 permutation that starts with "EDISW."

3. **Calculating the Total Rank:**
   - Permutations starting with "D": 60
   - Permutations starting with "ED": 24
   - Permutations starting with "EDI": 6
   - Permutations starting with "EDIS": 2
   - Permutations starting with "EDISW": 1

   Adding them all up, we get:

   Rank of "EDWISE" = 60 (from D) + 24 (from ED) + 6 (from EDI) + 2 (from EDIS) + 1 (from EDISW) + 1 (EDWISE itself)

   **Rank of "EDWISE" = 94**

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### Do you have any questions, or would you like more details?

### Related Questions:

1. How do you find the rank of a word with repeated characters in permutation?
2. What is the rank of the word "EDWISE" if each letter could only be used once?
3. How can we generalize the method to find the rank of a word in a given set of permutations?
4. How do permutations with repeated letters differ from those with distinct letters?
5. Can the rank calculation be automated for larger words?

### Tip:  
When calculating the rank of a word, always ensure to account for repeated characters by dividing by the factorial of the count of those characters.

Learn how to calculate the rank of the word "EDWISE" in permutations using combinatorial principles and factorial calculations. Follow a step-by-step explanation to determine the position of "EDWISE" in lexicographical order. Suitable for advanced high school students and anyone interested in combinatorial mathematics.

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