Unit 2 - Permutations and Organized Counting Handout Unit 2 Outline Learning Goals Warm-up activity Note - Fundamental Counting Principle or multiplicative rule Note - Indirect Method Note - Additive Counting Principle or Rule of sum

Organized Counting Learning Goals I can draw tree diagrams to model counting problems. I can use the fundamental counting principle (multiplicative principle) additive counting principle (rule of sum) and indirect method to solve counting problems.

Methods of Counting Solution 1 Warm - Up NDSS is holding their annual student council elections. There are 3 candidates for President, 3 for Vice-President, 5 for Social Co-Ordinator and 2 for Communications. Assuming there are no running-mates (every position is chosen independent of others) how many different councils are possible? Solution 1

Methods of Counting Solution 2 Warm - Up NDSS is holding their annual student council elections. There are 3 candidates for President, 3 for Vice-President, 5 for Social Co-Ordinator and 2 for Communications. Assuming there are no running-mates (every position is chosen independent of others) how many different councils are possible? Solution 2

Methods of Counting - Tree diagrams Combinatronics "branch of mathematics dealing with ideas and methods for counting" Textbook page 225 Example 1 Chez-Piggy restaurant offers a dinner special where you can choose 1 appetizer, 1 entree and 1 dessert off a special menu. Appetizers include a spinach salad, frites or soup. Entree's include fish, chicken or steak. Desserts include chocolate mousse, pavlova, and fresh fruit. What is the total number of meal possibilities for the special menu? Solution Given - Required - Solution -

Methods of Counting - Tree diagrams Example 1 - Tree Diagram solution Chez-Piggy restaurant offers a dinner special where you can choose 1 appetizer, 1 entree and 1 dessert off a special menu. Appetizers include a spinach salad, frites or soup. Entree's include fish, chicken or steak. Desserts include chocolate mousse, pavlova, and fresh fruit. What is the total number of meal possibilities for the special menu? Solution chocolate mousse fresh fruit pavlova fish chicken steak spinach salad frites soup

Methods of Counting - MCP Example 1 - Product Rule solution Multiplicative Counting Principle (fundamental counting principle) Chez-Piggy restaurant offers a dinner special where you can choose 1 appetizer, 1 entree and 1 dessert off a special menu. Appetizers include a spinach salad, frites or soup. Entree's include fish, chicken or steak. Desserts include chocolate mousse, pavlova, and fresh fruit. What is the total number of meal possibilities for the special menu? Solution Given - Required - Solution - Total # =

Methods of Counting - MCP Multiplicative Counting Principle (fundamental counting principle) The multiplicative counting principle is used when a problem has several stages with separate choices. The total number of choices is m x n x p... m = total number of first stage choices n = total number of second stage choices p = total number of third stage choices and so on

Methods of Counting - Rule of Sum Example 2 How many ways are there of selecting either a red face card or a black ace from a standard deck of cards? Solution Given - Required - Solution -

ways in which one of these actions can occur Methods of Counting - Rule of Sum Additive Counting Principle or Rule of Sum If one mutually exclusive action can occur in m ways, and a second in n ways a third in p ways and so on, then there are m + p + n ways in which one of these actions can occur

Methods of Counting - Indirect Method Example 3 How many 5 digit locker numbers can be created if at least one of the numbers is repeated? Given - Lockers with 5 numbers = Lockers with non-repeats Required - Lockers with at least 1 repeat Solution -