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Learn permutation of all objects using factorial formulas. Arrange all objects with worked examples and practical exercises.
---
## Understanding Permutation of All Objects
**Permutation of all objects** is an arrangement of all available objects, namely $$n$$ items from $$n$$ objects, where each object is used exactly once and **the order of arrangement is very important**. In this case, we use all available objects without any remaining.
Visible text: **Permutation of all objects** is an arrangement of all available objects, namely items from objects, where each object is used exactly once and **the order of arrangement is very important**. In this case, we use all available objects without any remaining.
Imagine arranging a photo line for $$5 \text{ students}$$. All five students are placed in one line. Each student gets one position, none sits on the backup bench, and Andi's position in front or back gives different results.
Visible text: Imagine arranging a photo line for . All five students are placed in one line. Each student gets one position, none sits on the backup bench, and Andi's position in front or back gives different results.
## Complete Permutation Formula
For permutation of $$n$$ items from $$n$$ objects, **the formula used** is:
Visible text: For permutation of items from objects, **the formula used** is:
Component: MathContainer
Children:
```math
nP_n = \frac{n!}{(n-n)!} = \frac{n!}{0!}
```
**Explanation of why this formula becomes simple:**
- $$(n-n) = 0$$, so the denominator becomes $$0!$$
- Based on mathematical definition, $$0! = 1$$
- Therefore: $$\frac{n!}{0!} = \frac{n!}{1} = n!$$
Visible text: - , so the denominator becomes
- Based on mathematical definition,
- Therefore:
So, the formula is:
Component: MathContainer
Children:
```math
nP_n = n!
```
## Factorial Concept
**Factorial** is the consecutive multiplication of positive integers. The factorial of a number $$n$$ is written as $$n!$$ and defined as:
Visible text: **Factorial** is the consecutive multiplication of positive integers. The factorial of a number is written as and defined as:
Component: MathContainer
Children:
```math
n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1
```
**Special rules** for factorial:
- $$0! = 1$$ (based on mathematical definition)
- $$1! = 1$$
Visible text: - (based on mathematical definition)
-
## Application in Daily Life
### School Organization
Suppose there are $$4 \text{ students}$$ who will fill $$4$$ positions in a student committee: president, vice president, secretary, and treasurer. Each student can only hold one position.
Visible text: Suppose there are who will fill positions in a student committee: president, vice president, secretary, and treasurer. Each student can only hold one position.
The number of ways to arrange the leadership is:
Component: MathContainer
Children:
```math
4P_4 = 4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways}
```
### Seating Arrangement
A family consisting of $$6$$ members will sit in a row on a sofa for a family photo. The number of ways they can be arranged is:
Visible text: A family consisting of members will sit in a row on a sofa for a family photo. The number of ways they can be arranged is:
Component: MathContainer
Children:
```math
6P_6 = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \text{ ways}
```
## Systematic Calculation Steps
To calculate permutation of $$n$$ items from $$n$$ objects, follow these steps:
Visible text: To calculate permutation of items from objects, follow these steps:
1. **Identify the number of objects**: Ensure all objects will be used
2. **Apply the formula**: Use $$n!$$
3. **Calculate factorial**: Multiply consecutively from $$n$$ to $$1$$
4. **Verify the result**: Ensure the calculation is correct
Visible text: 1. **Identify the number of objects**: Ensure all objects will be used
2. **Apply the formula**: Use
3. **Calculate factorial**: Multiply consecutively from to
4. **Verify the result**: Ensure the calculation is correct
### Detailed Calculation
Calculating $$5!$$ with clear steps:
Visible text: Calculating with clear steps:
Component: MathContainer
Children:
```math
5! = 5 \times 4 \times 3 \times 2 \times 1
```
```math
= 20 \times 3 \times 2 \times 1
```
```math
= 60 \times 2 \times 1
```
```math
= 120 \times 1 = 120
```
## Difference from Partial Permutation
**Complete permutation ($$n$$ from $$n$$ objects)**: Uses all available objects.
**Partial permutation ($$r$$ from $$n$$ objects)**: Only uses some objects.
Visible text: **Complete permutation ( from objects)**: Uses all available objects.
**Partial permutation ( from objects)**: Only uses some objects.
**Concrete example**:
- Complete permutation: Arranging $$5 \text{ books}$$ in $$5 \text{ positions}$$ on a shelf is $$5! = 120 \text{ ways}$$
- Partial permutation: Selecting and arranging $$3 \text{ books}$$ from $$5 \text{ available books}$$ is $$5P_3 = 60 \text{ ways}$$
Visible text: - Complete permutation: Arranging in on a shelf is
- Partial permutation: Selecting and arranging from is
**Detailed calculation for partial permutation**:
Component: MathContainer
Children:
```math
5P_3 = \frac{5!}{(5-3)!} = \frac{5!}{2!}
```
```math
= \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1}
```
```math
= \frac{120}{2} = 60 \text{ ways}
```
In permutation of $$n$$ items from $$n$$ objects, **no objects are left over** and **all positions must be filled**.
Visible text: In permutation of items from objects, **no objects are left over** and **all positions must be filled**.
## Exercises
1. A photography team wants to arrange $$7 \text{ models}$$ for a photo session in one line. How many different ways can they arrange the seven models?
2. In a running competition, there are $$5 \text{ participants}$$ who must all finish. How many different finishing order possibilities are there?
3. A chef wants to arrange $$6 \text{ different types of food}$$ on a table in one straight line. How many different ways can he arrange the food?
4. A library has $$8 \text{ different books}$$ that will be arranged on one shelf. If all books must be placed on that shelf, how many possible arrangements are there?
Visible text: 1. A photography team wants to arrange for a photo session in one line. How many different ways can they arrange the seven models?
2. In a running competition, there are who must all finish. How many different finishing order possibilities are there?
3. A chef wants to arrange on a table in one straight line. How many different ways can he arrange the food?
4. A library has that will be arranged on one shelf. If all books must be placed on that shelf, how many possible arrangements are there?
### Answer Key
1. Given: $$7 \text{ models}$$ will be arranged in one line (all models used)
```math
7P_7 = 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \text{ ways}
```
2. Given: $$5 \text{ running participants}$$ with different finishing order (all participants finish)
```math
5P_5 = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \text{ possible orders}
```
3. Given: $$6 \text{ types of food}$$ will be arranged in one straight line (all food arranged)
```math
6P_6 = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \text{ ways}
```
4. Given: $$8 \text{ different books}$$ will be arranged on one shelf (all books arranged)
```math
8P_8 = 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \text{ ways}
```
Visible text: 1. Given: will be arranged in one line (all models used)
2. Given: with different finishing order (all participants finish)
3. Given: will be arranged in one straight line (all food arranged)
4. Given: will be arranged on one shelf (all books arranged)