Counting Principles and Tree Diagrams

Fundamental counting principle The Fundamental Counting Principle states that if there are x ways to choose a first item and y ways to choose a second item, then there are x(y) ways to choose all items.

For example A telephone company is assigned a new area code and can issue new 7- digit phone numbers. All phone numbers are equally likely. Find the number of possible 7-digit phone numbers Use the Fundamental Counting Principle: 1st digit 2nd digit 3rd digit 4th digit 5th digit 6th digit 7th digit ? ? ? ? ? ? ? 10 10 10 10 10 10 10 There are 10 choices for each digit (0-9), so there are 10(10)(10)(10)(10)(10)(10) = 10,000,000 phone number options

You try A telephone company is assigned a new area code and can issue new 7- digit phone numbers. All phone numbers are equally likely. Find the probability of a phone number that does not contain an 8. First, use the fundamental counting principle to find the number of phone numbers that do not contain an 8. 9(9)(9)(9)(9)(9)(9) = 4,782,969 P(no 8) = 4,782,969 = .478 10,000,000

Using a tree diagram The Fundamental Counting Principle tells you only the number of outcomes in some experiments, not what the outcomes are. A tree diagram is a way to show all possible outcomes. For example: A bakery sells two famous cookies, chocolate chip and oatmeal, and they sell these cookies with two types of milk, white or chocolate. The table below shows all combinations.

Tree diagrams Using the Fundamental Counting Principle, we know that we should have 4 options: 2 milk types and 2 cookies …2(2)=4. The tree diagram shows us the options rather than just giving a total.

You try You are going on a trip. You can pack 2 pairs of pants, 3 shirts, and 2 sweaters for your vacation. Use a tree diagram to show all outfit options you can make if each outfit consists of a pair of pants, a shirt, and a sweater. There are 12 total outfits to choose from 2(3)(2) = 12

The addition counting principle If one group contains x objects and a second group contains y objects, and the groups have no objects in common, then there are x + y options. For example How many items can you choose from Bergen’s Deli menu? None of the lists contains identical items, so use the Addition Counting Principle. Total Choices = Sandwiches + Salads + Soups T = 4 + 3 + 3 There are 10 total items to choose from Sandwiches Salads Soups Turkey Ham Roast Beef Reuben Cobb Salad Taco Salad Grill Chicken Salad Tomato Chicken Noodle French Onion