Sample Spaces, Tree Diagrams, and the Counting Principle

How many different 5-digit zip codes are there if digits can be repeated but it can't start with a zero?

How many different 5-digit zip codes are there if digits can’t be repeated?

You are buying a car and can choose from 2 different models and 4 different colors.

How many different cars could you purchase?

• 2

• 4

• 6

• 8

• 16

You visit the ice cream shop on vacation.

Your flavor choices are vanilla or chocolate; the toppings are sprinkles, nuts, or cherries; and the sauces are caramel, whipped cream, or chocolate syrup.

How many different ice cream sundaes exist consisting of one flavor of ice cream, one topping and one sauce?

• 6

• 9

• 18

• 27

• 12

Your email password consists of three lower case letters followed by 4 digits (0 to 9).

How many different passwords exist?

• 17,576

• 456,976

• 676,000

• 4,569,760,000

• 175,760,000

How many 6-character license plates are there if you can use all 26 letters and all 10 digits (0 to 9)?

• 2,176,782,336

• 19,770,609,664

• 141,167,095,653

• 308,915,776

• 360,000,000,000

You are creating a password that consists of 5 different letters of the alphabet where each letter can only be used once.

How many different passwords are there?

• 120

• 3024

• 26,400

• 15,600

• 7,893,600

Daisy is choosing in the market for a new car. After being pre-approved for a loan, she heads to the dealership and the dealer asks her what kind of car she would like. Daisy can choose from the following options: - Type: Sedan or SUV - Color: Red, Blue, Black, White, or Green - Accessories: Basic Package, Advanced Package, or Premium Package - Windows: Tinted or Not Tinted Which counting method should you use to calculate the total number of cars that Daisy could buy if she chooses one Type, one Color, one Accessory package, and one Window option? Calculate the total number of unique cars that Daisy could buy.

• Fundamental Counting Principle; 60 total unique cars

• Permutations; 151,200 total unique cars

• Combinations; 210 total unique cars

• Permutations; 870 total unique cars

• Combinations; 435 total unique cars

Which counting method should you use to calculate the total number of ways that 5 students can be chosen to present today? Calculate the total number of groups of 5 students that could present today.

• Fundamental Counting Principle; 150 different ways

• Permutations; 17,100,720 different ways

• Combinations; 142,506 different ways

• Permutations; 6,375,600 different ways

• Combinations; 53,130 different ways

Once Sidar makes her insurance choices, she has to schedule a meeting with a Human Resources specialist. They have 14 time slots available and she is told to list her top three preferred meeting times in order from most preferred to least preferred. Which counting method should be used to calculate how many "Top 3s" Sidar could write down? Calculate how many "Top 3" preferred meeting times Sidar could write down.

• Fundamental Counting Principle; 42 different "Top 3"s

• Permutations; 2,184 different "Top 3"s

• Combinations; 364 "Top 3"s

• Permutations; 14,529,715,200 different "Top 3"s

• Combinations; 165 different "Top 3"s