Unit 2 - Permutations and Organized Counting

Handout Unit 2 OutlineLearning GoalsWarm-up activityNote - Fundamental Counting Principle or multiplicative ruleNote - Indirect MethodNote - Additive Counting Principle or Rule of sum

Learning GoalsI 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.

Organized Counting

Warm - UpNDSS 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?

Methods of Counting

Solution 1

Warm - UpNDSS 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?

Methods of Counting

Solution 2

Combinatronics"branch of mathematics dealing with ideas and methods for counting"Textbook page 225

Methods of Counting - Tree diagrams

Example 1Chez-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 solutionChez-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?

Solutionchocolate mousse

fresh fruit

pavlova fish

chicken

steak

chocolate mousse

fresh fruit

pavlova

chocolate mousse

fresh fruit

pavlova

chocolate mousse

fresh fruit

pavlova fish

chicken

steak

chocolate mousse

fresh fruit

pavlova

chocolate mousse

fresh fruit

pavlova

chocolate mousse

fresh fruit

pavlova fish

chicken

steak

chocolate mousse

fresh fruit

pavlova

chocolate mousse

fresh fruit

pavlova

spinachsalad

frites

soup

Methods of Counting - MCP

Example 1 - Product Rule solutionMultiplicative 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 choicesn = total number of second stage choicesp = total number of third stage choicesand so on

Methods of Counting - Rule of Sum

Example 2How 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 -

Methods of Counting - Rule of Sum

Additive Counting Principle or Rule of SumIf 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 3How 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 -