Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Define permutations and combinations.
• Solve real-life problems involving permutations and combinations.
• Apply these concepts to make predictions with data.
• Work together to solve problems.

Lesson summary

In our daily lives, we often need to figure out the number of ways something can be done. For example, how many different ways can the Black Stars starting eleven be arranged on the field? How many different combinations of subjects can you choose for your WASSCE exams? This lesson introduces two powerful mathematical tools, permutations and combinations, that help us count these possibilities accurately without listing them all. Understanding these concepts is crucial for making informed decisions and predictions in fields like business, science, and even sports.

Lesson notes

A. The Fundamental Counting Principle (Recap)

This is the foundation for our topic. It states that if one task can be done in `m` ways, and a second task can be done in `n` ways, then both tasks can be done in `m × n` ways. Example: If you have 3 shirts (red, blue, green) and 2 pairs of trousers (black, khaki), the total number of different outfits you can wear is 3 × 2 = 6 outfits. B. Factorial Notation (!)

The factorial of a non-negative integer 'n', denoted by `n!`, is the product of all positive integers less than or equal to n. `n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1` Examples: `5! = 5 × 4 × 3 × 2 × 1 = 120` `3! = 3 × 2 × 1 = 6` Special Case: By definition, `0! = 1`. C. Permutations (Arrangements where Order Matters)

A permutation is an arrangement of a set of objects in a specific order. Think of it as Permutation = Position. If the order of the objects changes, you get a new permutation. Permutation of 'n' distinct objects: The number of ways to arrange 'n' different objects is `n!`. Example: How many ways can the letters A, B, C be arranged? Solution: We have 3 distinct letters, so n=3. The number of arrangements is `3! = 3 × 2 × 1 = 6`. (The arrangements are ABC, ACB, BAC, BCA, CAB, CBA). Permutation of 'r' objects from 'n' distinct objects: The number of ways to arrange 'r' objects selected from a set of 'n' objects is given by the formula: `nPr = n! / (n-r)!`

Evaluation guide

• Solve real life problems involving permutations.
• Group work, Talk for Learning, and Building on what others say.
• Permutation
• Learners in groups recall the concepts of arrangements or permutation as arranging objects where