Download Organized Counting - Ms. Pasic

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
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 runningmates (every position is chosen independent of others) how many different councils are possible?
Solution 1
Methods of Counting
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 runningmates (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?
So
lution
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?
So
lution
chocolate mousse
fish
pavlova
fresh fruit
spinach
salad
chocolate mousse
chicken
pavlova
fresh fruit
chocolate mousse
steak
pavlova
fresh fruit
chocolate mousse
fish
pavlova
fresh fruit
chocolate mousse
frites
chicken
pavlova
fresh fruit
chocolate mousse
steak
pavlova
fresh fruit
chocolate mousse
fish
pavlova
fresh fruit
chocolate mousse
soup
chicken
pavlova
fresh fruit
chocolate mousse
steak
pavlova
fresh fruit
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?
S
olution
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?
S
olution
Given Required Solution -
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 -