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Learn slot filling rule in combinatorics with table, tree diagram, and multiplication methods. Learn counting arrangements with examples and exercises.
---
## Understanding of Slot Filling Rule
**Slot filling rule** is a method to determine the number of ways to place objects in available slots. This concept is very useful in solving combinatorial problems where we need to count all possible arrangements or choices that can be made.
Imagine filling out a form that has several columns. Each column has certain options, and we want to know how many different ways there are to fill out the entire form.
## Table Rule Method
**Table method** presents all possible combinations in a systematic table format. Each row and column represents choices from different categories.
Suppose a student wants to choose an online learning package. There are three platforms (Platform A, Platform B, Platform C) and four subjects (Mathematics, Physics, Chemistry, Biology).
Using a table, we can see all possible combinations:
| Platform | Mathematics | Physics | Chemistry | Biology |
|----------|-------------|---------|-----------|---------|
| Platform A | A-Math | A-Phys | A-Chem | A-Bio |
| Platform B | B-Math | B-Phys | B-Chem | B-Bio |
| Platform C | C-Math | C-Phys | C-Chem | C-Bio |
Component: MathContainer
Children:
```math
\text{Total combinations} = 3 \times 4 = 12 \text{ ways}
```
## Tree Diagram Method
**Tree diagram** depicts each choice as a tree branch. This method helps visualize step-by-step decision making.
For the same case, the tree diagram starts from one initial point, then branches into available choices.
1. Level $$1$$:
```math
1 \text{ initial point} \rightarrow 3 \text{ platform branches}
```
2. Level $$2$$:
```math
3 \text{ platforms} \rightarrow 3 \times 4 = 12 \text{ subject branches}
```
3. Branch structure:
```math
\text{Start} \rightarrow \text{Platform} \rightarrow \text{Subject}
```
4. Total complete routes:
```math
3 \times 4 = 12 \text{ combinations}
```
Visible text: 1. Level :
2. Level :
3. Branch structure:
4. Total complete routes:
## Multiplication Rule Method
**Multiplication rule** is the most efficient method to calculate the number of ways to fill available slots. If there are $$n$$ slots with each slot $$i$$ having $$k_i$$ choices, then the **total number of ways to fill** is:
Visible text: **Multiplication rule** is the most efficient method to calculate the number of ways to fill available slots. If there are slots with each slot having choices, then the **total number of ways to fill** is:
Component: MathContainer
Children:
```math
\text{Total ways} = k_1 \times k_2 \times k_3 \times \cdots \times k_n
```
### Example of Multiplication Rule Usage
A school wants to create access codes for digital learning systems. The code consists of:
- First slot: $$3 \text{ letters}$$ ($$A, B, C$$)
- Second slot: $$5 \text{ numbers}$$ ($$1, 2, 3, 4, 5$$)
- Third slot: $$2 \text{ symbols}$$ ($$@, \#$$)
Visible text: - First slot: ()
- Second slot: ()
- Third slot: ()
Then, the total different codes that can be created are:
Component: MathContainer
Children:
```math
k_1 = 3 \text{ (letter choices)}
```
```math
k_2 = 5 \text{ (number choices)}
```
```math
k_3 = 2 \text{ (symbol choices)}
```
```math
\text{Total different codes} = 3 \times 5 \times 2 = 30 \text{ codes}
```
## Cases with Restrictions
In some situations, there are **certain restrictions** that affect the number of choices in each slot.
### Repetition Not Allowed
If the same object **cannot be used repeatedly**, then each filled slot will reduce the choices for the next slot.
**Example:** Creating a $$3 \text{-digit}$$ number from digits $$2, 3, 4, 5, 6$$ without repetition.
Visible text: **Example:** Creating a number from digits without repetition.
Component: MathContainer
Children:
```math
\text{First slot} = 5 \text{ choices}
```
```math
\text{Second slot} = 4 \text{ choices (1 digit already used)}
```
```math
\text{Third slot} = 3 \text{ choices (2 digits already used)}
```
```math
\text{Total numbers} = 5 \times 4 \times 3 = 60 \text{ numbers}
```
### Repetition Allowed
If the same object **can be used repeatedly**, then the choices in each slot remain the same.
For the same case with repetition allowed:
Component: MathContainer
Children:
```math
\text{Each slot} = 5 \text{ choices}
```
```math
\text{Total numbers} = 5 \times 5 \times 5 = 125 \text{ numbers}
```
## Exercises
1. An electronics store sells smartphones with $$4$$ different brands, each available in $$3$$ memory capacities and $$5$$ color choices. How many different smartphone combinations are there?
2. To create a password consisting of $$1$$ letter followed by $$2$$ numbers, where the letter is chosen from A, B, C, D and numbers are chosen from $$\{1, 2, 3, 4, 5\}$$ without repetition. How many passwords can be created?
3. From city $$P$$ to city $$R$$ through city $$Q$$, there are $$3$$ roads from $$P$$ to $$Q$$ and $$4$$ roads from $$Q$$ to $$R$$. How many different routes can be chosen for the journey from $$P$$ to $$R$$?
4. Creating license plate numbers consisting of $$2$$ letters followed by $$3$$ numbers. If there are $$26$$ letters and $$10$$ numbers ($$0\text{-}9$$) available, and repetition is allowed, how many license plate numbers can be created?
Visible text: 1. An electronics store sells smartphones with different brands, each available in memory capacities and color choices. How many different smartphone combinations are there?
2. To create a password consisting of letter followed by numbers, where the letter is chosen from A, B, C, D and numbers are chosen from without repetition. How many passwords can be created?
3. From city to city through city , there are roads from to and roads from to . How many different routes can be chosen for the journey from to ?
4. Creating license plate numbers consisting of letters followed by numbers. If there are letters and numbers () available, and repetition is allowed, how many license plate numbers can be created?
### Answer Key
1. Given: $$4 \text{ brands}$$, $$3 \text{ memory capacities}$$, $$5 \text{ color choices}$$
```math
\text{Total combinations} = 4 \times 3 \times 5 = 60 \text{ smartphone combinations}
```
2. Given: $$1 \text{ letter}$$ from $$\{A, B, C, D\}$$, $$2 \text{ numbers}$$ from $$\{1, 2, 3, 4, 5\}$$ without repetition
```math
\text{Letter choices} = 4
```
```math
\text{First number choices} = 5
```
```math
\text{Second number choices} = 4 \text{ (without repetition)}
```
```math
\text{Total passwords} = 4 \times 5 \times 4 = 80 \text{ passwords}
```
3. Given: $$3 \text{ roads}$$ from $$P$$ to $$Q$$, $$4 \text{ roads}$$ from $$Q$$ to $$R$$
```math
\text{Total routes} = 3 \times 4 = 12 \text{ different routes}
```
4. Given: $$2 \text{ letters}$$ from $$26 \text{ letters}$$, $$3 \text{ numbers}$$ from $$10 \text{ numbers}$$, repetition allowed
```math
\text{First letter choices} = 26
```
```math
\text{Second letter choices} = 26
```
```math
\text{Each number choices} = 10
```
```math
\text{Total license plates} = 26 \times 26 \times 10 \times 10 \times 10
```
```math
= 676 \times 1000 = 676000 \text{ license plates}
```
Visible text: 1. Given: , ,
2. Given: from , from without repetition
3. Given: from to , from to
4. Given: from , from , repetition allowed