The Counting Principle Advanced Math Topics

Vocabulary Independent Events: choice of one thing does NOT affect the choice of another Independent Events: choice of one thing does NOT affect the choice of another Dependent Events: choice of one thing does affect the choice of another Dependent Events: choice of one thing does affect the choice of another Outcome: the result of a single trial Outcome: the result of a single trial Sample Space: set of all possible outcomes Sample Space: set of all possible outcomes Event: one or more outcomes of a trial Event: one or more outcomes of a trial

The Fundamental Counting Principle If event M can occur m ways and event N can occur in n ways, then event M followed by event N can occur in mn ways If event M can occur m ways and event N can occur in n ways, then event M followed by event N can occur in mn ways Basically multiply the number of ways for each event to get the total number of ways the events can occur Basically multiply the number of ways for each event to get the total number of ways the events can occur

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

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

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

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

*in the blank box of the definitions Dependent: Dependent: If you need to schedule 4 appointments and they are all available in the same four times, how many different appointment schedules could you make? If you need to schedule 4 appointments and they are all available in the same four times, how many different appointment schedules could you make?

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

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

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

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

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

Examples Charlita wants to take 6 different classes next year. Assuming that each class is offered each period, how many different schedules could she have? Charlita wants to take 6 different classes next year. Assuming that each class is offered each period, how many different schedules could she have? Note if she takes Algebra II first period she won’t take it another period… so this is a dependent event. Note if she takes Algebra II first period she won’t take it another period… so this is a dependent event.

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

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