Kaidah Pencacahan 3 - Permutasi dan Kombinasi Matematika Wajib Kelas 12

Introduction

This tutorial covers the essential concepts of permutations and combinations, which are foundational elements in counting principles necessary for understanding probability. By mastering these concepts, you'll be better prepared for advanced mathematical topics.

Step 1: Understanding Permutations

Permutations refer to the different arrangements of a set of items where the order matters. Here's how to calculate permutations:

• Formula: The number of permutations of n items taken r at a time is given by:

  [ P(n, r) = \frac{n!}{(n-r)!} ]

• Definitions:

  • n! (n factorial): The product of all positive integers up to n.
  • r: The number of items to arrange.

• Example

  If you have 5 books and want to know how many ways you can arrange 3 of them:

  [ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60 ]

Step 2: Understanding Combinations

Combinations involve selecting items from a group where the order does not matter. To calculate combinations, follow these steps:

• Formula: The number of combinations of n items taken r at a time is given by:

  [ C(n, r) = \frac{n!}{r!(n-r)!} ]

• Definitions:

  • r!: The factorial of r.

• Example

  If you want to choose 3 books from a set of 5:

  [ C(5, 3) = \frac{5!}{3! \cdot 2!} = \frac{120}{6 \cdot 2} = 10 ]

Step 3: Key Differences between Permutations and Combinations

Understanding the distinction between permutations and combinations is crucial:

• Permutations:

  • Order is important.
  • Used when arrangements are required.

• Combinations:

  • Order is not important.
  • Used when selections are made.

Step 4: Practical Applications

Both permutations and combinations have real-world applications, such as:

• Event Planning: Determining seating arrangements (permutations) or forming committees (combinations).
• Probability: Calculating outcomes in games or experiments.

Conclusion

By grasping the concepts of permutations and combinations, you will enhance your mathematical skills and prepare yourself for more complex topics in probability. Practice these formulas with various examples to solidify your understanding. For further learning, consider exploring additional resources on counting principles and probability theory.