VIPSolutionsIN STEP 1 GIVE THE INTRODUCTION OF THE CONCEPT AND GIVE ANSWER FOR EACH PART OF THE QUESTION IN EACH DIFFERENT STEP WITH CLEAR EXPLANATION AND IN THE FINAL STEP GIVE THE WHOLE FINAL ANSWER IN JUST VERY FEW SENTENCES AND MOREOVER I NEED COMPLETE AND CLEAR ANSWER at last explain what we did in each step in just one sentence7: Determine the umber of permutations of {1,2,3,4,5,6,7,8) in which mo even integer is mata) position. Foe cxaunpl, 3258176 i god, but 428176 is id because of the position of the 2.
Question:
IN STEP 1 GIVE THE INTRODUCTION OF THE CONCEPT AND GIVE ANSWER FOR EACH PART OF THE QUESTION IN EACH DIFFERENT STEP WITH CLEAR EXPLANATION AND IN THE FINAL STEP GIVE THE WHOLE FINAL ANSWER IN JUST VERY FEW SENTENCES AND MOREOVER I NEED COMPLETE AND CLEAR ANSWER
at last explain what we did in each step in just one sentence
7: Determine the umber of permutations of {1,2,3,4,5,6,7,8) in which mo even integer
is mata) position. Foe cxaunpl, 3258176 i god, but 428176 is id because of the
position of the 2.
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Asked by: Test User 100081
Created at: 2025-05-07 05:48:05
Asked by: Test User 100081
Created at: 2025-05-07 05:48:05
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---
### **STEP 1: Introduction to the Concept**
This problem asks us to count the number of permutations of the set {1,2,3,4,5,6,7,8} such that **no even integer appears in its natural position** (i.e., for every even number \(2k\), it cannot be in position \(2k\)). This is a classical example of a *restricted permutation* problem, specifically called a "derangement" (or subcase of it).
---
### **STEP 2: Identify the Even Numbers and Their Positions**
The even numbers in the set {1,2,3,4,5,6,7,8} are {2, 4, 6, 8}, located at positions 2, 4, 6, 8 (if we count positions from 1 to 8).
**Restriction:** None of these even numbers can be in their corresponding positions.
---
### **STEP 3: Permuting the Even Numbers (Derangement)**
We need to arrange the even numbers {2,4,6,8} in the positions {2,4,6,8} **such that none is in its own position**.
This is the definition of a **derangement** for 4 objects, usually denoted as \( !4 \).
The number of derangements for \( n \) objects is:
\[
!n = n! \left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \ldots + (-1)^n \frac{1}{n!}\right)
\]
For \( n=4 \):
\[
!4 = 4! \left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}\right)
= 24 \left(1 - 1 + 0.5 - 0.1667 + 0.0417\right)
= 24 \times 0.375 = 9
\]
So, there are **9 ways** to place the even numbers.
---
### **STEP 4: Permuting the Odd Numbers**
There are no restrictions on the odd numbers {1,3,5,7} in positions {1,3,5,7}.
So, **all 4! = 24** permutations are allowed for the odd numbers.
---
### **STEP 5: Total Number of Permutations**
By the multiplication principle, the total number of valid permutations is:
\[
\text{Total} = (\text{ways to arrange evens}) \times (\text{ways to arrange odds}) = 9 \times 24 = 216
\]
---
### **FINAL STEP: Concise Final Answer**
**There are 216 permutations of {1,2,3,4,5,6,7,8} in which no even integer is in its natural position.**
---
### **What We Did in Each Step (One Sentence Each):**
1. Explained the concept of restricted permutations (derangements).
2. Identified the even numbers and their restricted positions.
3. Calculated the derangements for the even numbers.
4. Counted all possible orderings for the odd numbers.
5. Multiplied the two results to get the final answer.
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