The Counting PrincipleThe Counting Principle

Advanced Math TopicsAdvanced Math Topics

VocabularyVocabulary

Independent Events:Independent Events: choice of one thing choice of one thing does NOT affect the choice of anotherdoes NOT affect the choice of another

Dependent Events:Dependent Events: choice of one thing choice of one thing does affect the choice of anotherdoes affect the choice of another

Outcome:Outcome: the result of a single trial the result of a single trial Sample Space:Sample Space: set of all possible set of all possible

outcomesoutcomes Event:Event: one or more outcomes of a trial one or more outcomes of a trial

The Fundamental Counting The Fundamental Counting PrinciplePrinciple

If event M can occur m ways and If event M can occur m ways and event N can occur in n ways, then event N can occur in n ways, then event M followed by event N can event M followed by event N can occur in mn waysoccur in mn ways Basically multiply the number of ways Basically multiply the number of ways

for each event to get the total number for each event to get the total number of ways the events can occurof ways the events can occur

ExamplesExamples

A sandwich cart offers the choice of A sandwich cart offers the choice of hamburger, chicken or fish on plain hamburger, chicken or fish on plain or sesame bun. How many or sesame bun. How many combinations of meat and bun are combinations of meat and bun are possible?possible? Note: meat choice does not affect bun Note: meat choice does not affect bun

choice so these events are independentchoice so these events are independent

Example:Example:

An ice cream shop offers two types of An ice cream shop offers two types of cones or a bowl, 17 flavors of ice cones or a bowl, 17 flavors of ice cream, and 40 toppings. How many cream, and 40 toppings. How many combinations of one-scoop, one-combinations of one-scoop, one-topping sundaes can you order?topping sundaes can you order?

ExamplesExamples

Kim won a contest on the radio. The Kim won a contest on the radio. The prize was a restaurant gift certificate prize was a restaurant gift certificate and tickets to a sporting event. She can and tickets to a sporting event. She can select one of three restaurants and select one of three restaurants and tickets for football, baseball, tickets for football, baseball, basketball, or hockey game. How many basketball, or hockey game. How many different ways can she select a different ways can she select a restaurant followed by a sporting restaurant followed by a sporting event?event?

ExampleExample

A sandwich menu offers customers a A sandwich menu offers customers a choice of white, wheat, or rye bread choice of white, wheat, or rye bread with one spread chosen from butter, with one spread chosen from butter, mustard, or mayonaise. How many mustard, or mayonaise. How many bread and spread combinations are bread and spread combinations are possible?possible?

*in the blank box of the *in the blank box of the definitionsdefinitions

Dependent:Dependent: If you need to schedule 4 appointments If you need to schedule 4 appointments

and they are all available in the same and they are all available in the same four times, how many different four times, how many different appointment schedules could you make?appointment schedules could you make?

ExampleExample

The Murray’s are choosing a trip to The Murray’s are choosing a trip to the beach or the mountains. They the beach or the mountains. They can travel by car, train, or plane. can travel by car, train, or plane. How many ways can the family How many ways can the family select a trip followed by means of select a trip followed by means of transportation?transportation?

ExampleExample

How many answering machine codes How many answering machine codes are possible if the code is two digits?are possible if the code is two digits?

ExampleExample

How many license plates can be How many license plates can be made if the first three places must be made if the first three places must be letters and the last three must be letters and the last three must be numbers?numbers?

ExamplesExamples

How many area codes are possible if How many area codes are possible if each area code is 3-digits?each area code is 3-digits?

ExamplesExamples

How many ATM pin numbers are How many ATM pin numbers are there if each pin number is 4 there if each pin number is 4 characters long and each character characters long and each character could be a number or a letter?could be a number or a letter?

ExamplesExamples

Charlita wants to take 6 different Charlita wants to take 6 different classes next year. Assuming that classes next year. Assuming that each class is offered each period, each class is offered each period, how many different schedules could how many different schedules could she have?she have? Note if she takes Algebra II first period Note if she takes Algebra II first period

she won’t take it another period… so she won’t take it another period… so this is a dependent event.this is a dependent event.

ExamplesExamples

Each player in a board game uses Each player in a board game uses one of six different pieces. If four one of six different pieces. If four players play the game, how many players play the game, how many different ways could the players different ways could the players choose their game pieces?choose their game pieces?

ExamplesExamples

An ice cream shop offers a choice of An ice cream shop offers a choice of two types of cones and 15 flavors of two types of cones and 15 flavors of ice cream, and the choice of peanuts, ice cream, and the choice of peanuts, chocolate sprinkles, or crushed oreos chocolate sprinkles, or crushed oreos for toppings. How many different 1-for toppings. How many different 1-scoop, 1-topping cones can a scoop, 1-topping cones can a customer order?customer order?