rank of edwise in permutation and combination when arranged in a dictionary

To find the rank of the word "EDWISE" when arranged in a dictionary in alphabetical order, we'll calculate how many words would come before "EDWISE" if all permutations of its letters were listed alphabetically.

### Step 1: Arrange the letters alphabetically
The word "EDWISE" consists of the letters: E, D, W, I, S, E. Arranging them in alphabetical order gives:
- **D, E, E, I, S, W**

### Step 2: Determine the rank
We need to calculate how many words come before "EDWISE" when arranged alphabetically.

#### 1. **Fix the first letter**:
   - **First position:**
     - The first letter in "EDWISE" is **E**. We first calculate how many words start with each letter that comes before "E" alphabetically (i.e., **D**).
     - For **D***, we have the remaining letters E, E, I, S, W to arrange:
     \[
     \text{Number of words starting with "D"} = \frac{5!}{2!} = \frac{120}{2} = 60
     \]

#### 2. **Fix the first letter as "E"**:
   - **Second position:**
     - After fixing the first letter as **E**, the next letter in "EDWISE" is **D**.
     - For each possible second letter, calculate the number of permutations:
       - For **EE*** (next in alphabetical order after "D"), permute the letters D, I, S, W:
       \[
       \text{Number of words starting with "EE"} = 4! = 24
       \]
       - For **EI***, permute the letters D, E, S, W:
       \[
       \text{Number of words starting with "EI"} = 4! = 24
       \]
       - For **ES***, permute the letters D, E, I, W:
       \[
       \text{Number of words starting with "ES"} = 4! = 24
       \]
       - For **EW***, permute the letters D, E, I, S:
       \[
       \text{Number of words starting with "EW"} = 4! = 24
       \]

#### 3. **Fix the first two letters as "ED"**:
   - **Third position:**
     - Now consider "ED". The third letter in "EDWISE" is **W**. We must calculate how many permutations come before "W".
       - For **EDI***, permute the letters E, S, W:
       \[
       \text{Number of words starting with "EDI"} = 3! = 6
       \]
       - For **EDS***, permute the letters E, I, W:
       \[
       \text{Number of words starting with "EDS"} = 3! = 6
       \]

#### 4. **Fix the first three letters as "EDW"**:
   - **Fourth position:**
     - Now that the first three letters are "EDW", we consider the next letter, which is **I**. 
     - We permute the remaining letters E, S:
     \[
     \text{Number of words starting with "EDWI"} = 2! = 2
     \]
     - Now consider the next letter in alphabetical order:
     - **EDWISE** is the exact word, and it's the first word when "EDWIS" is fixed.
       - **1** word for "EDWISE".

### Rank Calculation:
- Summing up the possibilities:
\[
\text{Rank of EDWISE} = 60 + 24 + 24 + 24 + 6 + 6 + 2 + 1 = 147
\]

**Therefore, the rank of the word "EDWISE" is 147** in a dictionary arrangement.

Would you like more details or have any questions?

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Here are 5 related questions:
1. How is the rank of the word "NUMBER" calculated in a dictionary arrangement?
2. What is the rank of the word "ALPHABET" using the same method?
3. How does the presence of repeated letters affect the rank calculation?
4. What would be the rank of the word "SCIENCE" in its dictionary arrangement?
5. How do you calculate the rank of a word in the presence of multiple identical letters?

**Tip:** When calculating the rank of a word, always count how many permutations start with letters that precede the current letter alphabetically, and factor in the effect of any repeated letters.

Learn how to calculate the rank of the word "EDWISE" when arranged in alphabetical order. Understand the concept of permutations and combinations, factorial calculations, and sequential arrangement methods. Suitable for advanced high school students studying permutations and combinations.

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