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Combinatorics

Slot Filling Rule

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:

PlatformMathematicsPhysicsChemistryBiology
Platform AA-MathA-PhysA-ChemA-Bio
Platform BB-MathB-PhysB-ChemB-Bio
Platform CC-MathC-PhysC-ChemC-Bio
Total combinations=3×4=12 ways\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:

    1 initial point→3 platform branches1 \text{ initial point} \rightarrow 3 \text{ platform branches}

  2. Level 2:

    3 platforms→3×4=12 subject branches3 \text{ platforms} \rightarrow 3 \times 4 = 12 \text{ subject branches}

  3. Branch structure:

    Start→Platform→Subject\text{Start} \rightarrow \text{Platform} \rightarrow \text{Subject}

  4. Total complete routes:

    3×4=12 combinations3 \times 4 = 12 \text{ combinations}

Multiplication Rule Method

Multiplication rule is the most efficient method to calculate the number of ways to fill available slots. If there are nn slots with each slot ii having kik_i choices, then the total number of ways to fill is:

Total ways=k1×k2×k3×⋯×kn\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 letters (A, B, C)
  • Second slot: 5 numbers (1, 2, 3, 4, 5)
  • Third slot: 2 symbols (@, #)

Then, the total different codes that can be created are:

k1=3 (letter choices)k_1 = 3 \text{ (letter choices)}
k2=5 (number choices)k_2 = 5 \text{ (number choices)}
k3=2 (symbol choices)k_3 = 2 \text{ (symbol choices)}
Total different codes=3×5×2=30 codes\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-digit number from digits 2, 3, 4, 5, 6 without repetition.

First slot=5 choices\text{First slot} = 5 \text{ choices}
Second slot=4 choices (1 digit already used)\text{Second slot} = 4 \text{ choices (1 digit already used)}
Third slot=3 choices (2 digits already used)\text{Third slot} = 3 \text{ choices (2 digits already used)}
Total numbers=5×4×3=60 numbers\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:

Each slot=5 choices\text{Each slot} = 5 \text{ choices}
Total numbers=5×5×5=125 numbers\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 (0-9) available, and repetition is allowed, how many license plate numbers can be created?

Answer Key

  1. Given: 4 brands, 3 memory capacities, 5 color choices

    Total combinations=4×3×5=60 smartphone combinations\text{Total combinations} = 4 \times 3 \times 5 = 60 \text{ smartphone combinations}

  2. Given: 1 letter from {A,B,C,D}\{A, B, C, D\}, 2 numbers from {1,2,3,4,5}\{1, 2, 3, 4, 5\} without repetition

    Letter choices=4\text{Letter choices} = 4
    First number choices=5\text{First number choices} = 5
    Second number choices=4 (without repetition)\text{Second number choices} = 4 \text{ (without repetition)}
    Total passwords=4×5×4=80 passwords\text{Total passwords} = 4 \times 5 \times 4 = 80 \text{ passwords}

  3. Given: 3 roads from P to Q, 4 roads from Q to R

    Total routes=3×4=12 different routes\text{Total routes} = 3 \times 4 = 12 \text{ different routes}

  4. Given: 2 letters from 26 letters, 3 numbers from 10 numbers, repetition allowed

    First letter choices=26\text{First letter choices} = 26
    Second letter choices=26\text{Second letter choices} = 26
    Each number choices=10\text{Each number choices} = 10
    Total license plates=26×26×10×10×10\text{Total license plates} = 26 \times 26 \times 10 \times 10 \times 10
    =676×1000=676000 license plates= 676 \times 1000 = 676000 \text{ license plates}
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