rank of edwise in permutation and combination

The rank of the word "EDWISE" in a dictionary arranged alphabetically using permutations and combinations can be calculated as follows:

### 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 the number of words that can be formed with the above letters and come before "EDWISE" in the dictionary.

#### 1. **Fix the first letter**:
   - **First position:**
     - Starting with **D**: All words starting with "D" will come before any word starting with "E".
     - The number of such words is determined by fixing "D" and permuting the remaining letters (E, E, I, S, W):
     \[
     \text{Number of such words} = \frac{5!}{2!} = \frac{120}{2} = 60
     \]

#### 2. **Fix the first letter as E**:
   - **Second position:**
     - Now consider words starting with "E". The second letter can be either D, E, I, S, or W. We need to find the number of words starting with each of these letters:
       - For **EDE***, fixing "D" next (as "E" is already fixed), we have four letters (E, I, S, W) left.
       - The number of such words is:
       \[
       \text{Number of words starting with "E", then "D"} = \frac{4!}{1!} = 24
       \]
       - For **EE***, fixing "E" next, we permute the letters (D, I, S, W):
       \[
       \text{Number of words starting with "EE"} = 24
       \]
       - For **EI***, we permute the letters (D, E, S, W):
       \[
       \text{Number of words starting with "EI"} = 24
       \]
       - For **ES***, we permute the letters (D, E, I, W):
       \[
       \text{Number of words starting with "ES"} = 24
       \]
       - For **EW***, we permute the letters (D, E, I, S):
       \[
       \text{Number of words starting with "EW"} = 24
       \]

#### 3. **Fix the first two letters as "ED"**:
   - **Third position:**
     - Now, considering words starting with "ED":
       - Fix "D" first:
       - We now have the letters E, I, S, W left.
       - For **EDW***, we fix "W" and permute the remaining letters (E, I, S):
       \[
       \text{Number of words starting with "EDW"} = \frac{3!}{1!} = 6
       \]
       - "EDW" is the desired prefix, and the remaining letters "ISE" match the given order.

### Rank Calculation:
- Summing up the possibilities:
\[
\text{Rank of EDWISE} = 60 + 24 + 24 + 24 + 6 + 1 = 139
\]
**Therefore, the rank of the word "EDWISE" is 139.**

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you calculate the rank of the word "SILENT"?
2. What is the rank of "EDUCATION" in a similar dictionary arrangement?
3. How do you calculate the rank of a word with repeated letters?
4. What is the rank of "COMBINATION"?
5. How does changing the position of letters affect the rank of a word?

**Tip:** When calculating ranks, always consider the alphabetical order of the letters carefully and the impact of repeated letters on permutations.

Learn how to calculate the rank of the word "EDWISE" in a dictionary arranged alphabetically using permutations and combinations. Understand the application of permutations and combinations, factorial calculations, and sequential letter arrangements.

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