How to Count The hardest thing we do in math is to get organized

Take My Test Number a paper from 1 to 4. Write “T” for true or “F” for false after each number. There are no questions, but there will be answers.

How did you do? Test Key 1.T 2.T 3.F 4.F Class Results: 1.T 2.T 3.F 4.F Mark “C” for correct. Mark “W” for wrong.

What should we have expected? We could list the possibilities. All Correct One Wrong Two Wrong Three Wrong All Wrong CCCCWCCCCCWWCWWWWWW CWCCCWCWWCWW CCWCCWWCWWCW CCCWWWCCWWWC WCWC WCCW 14641

What should we have expected? Use a tree diagram Take My Test C C C C W W C W W C C W W C W W C C C W W C W W C C W W C W

What should we have expected? Use a tree diagram Take My Test C C C C W W C W W C C W W C W W C C C W W C W W C C W W C W

What should we have expected? Use a tree diagram Take My Test C C C C W W C W W C C W W C W W C C C W W C W W C C W W C W

What should we have expected? Use a tree diagram Take My Test C C C C W W C W W C C W W C W W C C C W W C W W C C W W C W

What should we have expected? Use a tree diagram Take My Test C C C C W W C W W C C W W C W W C C C W W C W W C C W W C W

What should we have expected? Use a tree diagram Take My Test C C C C W W C W W C C W W C W W C C C W W C W W C C W W C W All Correct One Wrong Two Wrong Three Wrong Four Wrong

What If We Took a 10-Question Test? What Should We Expect? Remember Pascal’s Triangle! Two Outcomes for each Question: Correct or Wrong 10 Questions with equally likely random outcomes.

What Does Pascal’s Triangle Show for 10 Questions? How many different outcomes are there? How many ways to get all questions correct? How many ways to miss only one question? How many ways to get two questions wrong? How many ways to get only two questions correct? If you do not study, and just guess, what are your chances of getting at least 7 correct?

Listing, Tree Diagrams, Questions and Reasoning You have a red, blue and green shirt. You have blue or brown slacks. You have a paisley, hounds tooth, Tartan or felted wool jacket. How many different outfits can you make? If you have wear either the Tartan or paisley jacket, then how many outfits are possible?

Listing, Tree Diagrams, Questions and Reasoning How many different 3-digit identification numbers can be made with the numbers 1, 2, 3, 4, and 5 if no digit repeats? How many different 3-digit identification numbers can be made with the numbers 1, 2, 3, 4, and 5 if repetition is allowed?

Listing, Tree Diagrams, Questions and Reasoning How many ways can we select a President, Vice President and Treasurer from this class? How many ways can we form a three person committee from this class?

Listing, Tree Diagrams, Questions and Reasoning How many different sums are there when throwing two dice? How many ways can you get a sum of seven when throwing two dice? How many ways can you get a sum of four when throwing two dice?

Fundamental Counting Principle If a series of tasks are performed or decisions are made, and …. the first can happen in a ways, the second in b ways, The third in c ways, Etc… then All the tasks can be done in a x b x c x …. ways

Fundamental Counting Principle How many ways can we select a President, Vice President and Treasurer from this class? How many ways can we form a three person committee from this class?

Fundamental Counting Principle How many ways can you get two face cards when drawing two cards from a deck? (No repeat—no replacement) How many ways can you get a sum of eight when throwing two dice? (repeat okay) How many different two card hands are possible in a deck of cards? (No jokers!)

Fundamental Counting Principle You are buying a triple deck ice cream cone with strawberry, chocolate or vanilla as possible flavors. The flavors can be repeated, or not, and two cones are considered different if the flavors are the same but occur in different order. How many different cones are possible that have at least one scoop of chocolate? How many different cones are possible that have at least two scoops of chocolate? How many are possible that have at least one scoop each of strawberry and vanilla?

Fundamental Counting Principle You are buying a triple deck ice cream cone with strawberry, chocolate or vanilla as possible flavors. The flavors can be repeated, or not, and two cones are considered different if the flavors are the same but occur in different order. How many different cones are possible that have at least one scoop of chocolate? How many different cones are possible that have at least two scoops of chocolate? How many are possible that have at least one scoop each of strawberry of vanilla?

Listing, Tree Diagrams, Pascal’s Triangle Questions and Reasoning If I live here: How many ways to get here:

Fundamental Counting Principle Section 13.1 Problem 6, 10, 14, 26, 27, 30, 32