SAJC 2021_H1 Math_P & C notes

SAJC 2021 JC1 H1 Mathematics Chapter 1: Permutation and Combinations 0 Chapter 1 (Statistics) : Permutations and Combinations (Teacher’s Copy) Objectives: At the end of the chapter, you should be able to (a) understand and use the addition and multiplication principles for counting; (b) understand the concepts of permutation Pn r and combination Cn r , and differentiate between permutations and combinations; (c) understand the concept of arrangements of distinct objects in a line including cases involving restriction, and know how to calculate them Content 1.1 Basic Counting Principles 1.1.1 Addition Principle 1.1.2 Multiplication Principle 1.2 Permutations 1.3 Combinations

SAJC 2021 JC1 H1 Mathematics Chapter 1: Permutation and Combinations 1 1.1 BASIC COUNTING PRINC IPLES In the process of solving a counting problem, there are two very simple but basic principles that we always apply. They are called the Addition Principle and Multiplication Principle. 1.1.1 The Addition Principle (OR) Suppose you want to buy an ice cream. You have 5 flavours to choose from: Chocolate OR strawberry OR vanilla OR oreo OR mint. How many choices do you have? Answer: 1 + 1 + 1 + 1 + 1 = 5 Chocolate OR strawberry OR vanilla OR oreo OR mint In general, if there exists k non-overlapping categories of ways to perform an operation and the first category can be done in n1 ways, the second in n2 ways, …., the kth category in nk ways, then the operation can be done in 12 ... kn n n   ways. 1.1.2 The Multiplication Principle (AND) Suppose you want to buy an ice cream on a cone. There are 5 ice cream flavours to choose from and there are 3 types of cones. How many different choices do you have? No. of ways to choose a cone = 3 No. of ways to choose a flavour = 5 Total number of ways an ice cream AND a cone = 3 × 5 = 15 In general, if one operation can be done in n1 ways, and ( when this has been done ), the second operation can be performed in n2 ways and so on for k operations, then the k successive operations can be performed in 12 ... kn n n   ways. Cone 1. Waffle 2. Biscuit 3. Cup Flavour 1. Chocolate 2. Vanilla 3. Strawberry 4. Oreo 5. Mint

SAJC 2021 JC1 H1 Mathematics Chapter 1: Permutation and Combinations 2 Example 1 Mary and John go to a restaurant that offers 5 types of pizza and 6 types of salads. Mary would like to have either a pizza or a salad. John would like to have a pizza and a salad. How many choices do Mary and John have respectively? Solution: Mary John Case 1 Chooses pizza: 5 ways Case 2 Chooses salad: + 6 ways No. of choices = 11 Stage 1 pizza Stage 2 salad No. of choices = 5 ways x 6 ways = 30 Exercise 1 A bookshelf holds 6 different Engli