2. Course: Math Lit. Aim: Counting Principle Aim: How do I count the ways? Do Now: Use , , or both to make the following statement true. {s, r, t} _____ {s, r, t}

3. Course: Math Lit. Aim: Counting Principle Combinatorics Combinatorics – the study of counting the different outcomes of some task. Ex. A coin is flipped. Two possible outcomes: set {H, T} Ex. A die is tossed. 6 possible outcomes: {1, 2, 3, 4, 5, 6} List and then count the number of different outcomes that are possible when one letter from the word Tennessee is chosen. {T, e, n, s} List and then count the number of different outcomes that are possible when one letter from the word Mississippi is chosen. {M, i, s, p} Counting by Forming a List

4. Course: Math Lit. Aim: Counting Principle Combinatorics Experiment – An activity with an observable outcome. Sample Space – set of possible outcome for an experiment. Event – one or more of the possible outcomes of an experiment. An event is a subset of the sample space. Ex. Flipping a coin resulting in H; rolling a 5 when a die is tossed. Choosing the letter T are all experiments and a subset of each respective sample space. Each of these experiments are single or simple experiments – a single outcome.

5. Course: Math Lit. Aim: Counting Principle Model Problem One number is chosen from the sample space S = {1, 2, 3, 4, 5, 6, 7, 8 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} List the elements in the following events. a.The number is even b.The number is divisible by 5. c.The number is prime. {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} {5, 10, 15, 20} {2, 3, 5, 7, 11, 13, 17, 19

6. Course: Math Lit. Aim: Counting Principle Counting by Making a Table Multi-stage or compound experiments – experiments with more than one stage. Ex. rolling two dice: one red, one green. 36 possible outcomes How many red-green dice tosses results in a sum of seven? 6

7. Course: Math Lit. Aim: Counting Principle Model Problem Two-digits numbers are formed from the digits 1, 3, and 8. Find the sample space and determine the number of elements in the sample space. 138 1 3 8 11 1318 31 3338 81 8388 {11, 13, 18, 31, 33, 38, 81, 83, 88}

8. Course: Math Lit. Aim: Counting Principle Tree Diagram Tree diagram – a method for organizing multi-staged or compound experiment. Ex. The options for today’s lunch are the following: Main Course:spaghetti, hamburger or hot dog Drink:milk or coke Dessert:ice cream, apple pie or chocolate cake Multi-stage experiment

9. Course: Math Lit. Aim: Counting Principle Spaghetti Hamburger Hotdog Main CourseDrink Milk Coke Dessert Ice Cream Apple Pie Chocolate Cake Ice Cream Apple Pie Chocolate Cake S M I S M A S M C S C I S C A S C C Ice Cream Apple Pie Chocolate Cake Ice Cream Apple Pie Chocolate Cake Ice Cream Apple Pie Chocolate Cake Ice Cream Apple Pie Chocolate Cake Milk Coke Milk Coke H M A Ht M I Ht M A Ht M C H M I H M C H C I H C A H C C Ht C I Ht C A Ht C C Sample Space Tree Diagram

10. Course: Math Lit. Aim: Counting Principle Model Problem A true/false test consists of 10 questions. Draw a tree diagram to show the number of ways to answer the first three questions. T F T F T F T F T F T F T F TTT TTF TFT TFF FTF FFT FFF

11. Course: Math Lit. Aim: Counting Principle MJ Petrides Outerbridge Crossing Great Adventure How many different ways will get us from MJ Petrides to Great Adventure? Tracing the different routes we find there are 6 different routes. Is there a shortcut method for finding how many different routes there are? 3 2 Fundamental Counting Principle 3 x 2 = 6

12. Course: Math Lit. Aim: Counting Principle To find the total number of possible outcomes in a sample space, multiply the number of choices for each stage or event... in other words... If event M can occur in m ways and is followed by event N that can occur in n ways, then the event M followed by event N can occur in m · n ways. Counting Principle 2 events: m · n 3 events: m · n · o 4 events: m · n · o · p 5 events: etc. Fundamental Counting Principle

13. Course: Math Lit. Aim: Counting Principle 18 xx= Spaghetti Hamburger Hotdog Main CourseDrink Milk Coke Dessert Ice Cream Apple Pie Chocolate Cake Ice Cream Apple Pie Chocolate Cake S M I S M A S M C S C I S C A S C C Ice Cream Apple Pie Chocolate Cake Ice Cream Apple Pie Chocolate Cake Ice Cream Apple Pie Chocolate Cake Ice Cream Apple Pie Chocolate Cake Milk Coke Milk Coke H M A Ht M I Ht M A Ht M C H M I H M C H C I H C A H C C Ht C I Ht C A Ht C C Sample Space 323

14. Course: Math Lit. Aim: Counting Principle Jamie has 3 skirts - 1 blue, 1 yellow, and 1 red. She has 4 blouses - 1 yellow, 1 white, 1 tan and 1 striped. How many skirt-blouse outfits can she choose? Blue Yellow Red yellow striped white tan yellow striped white tan yellow striped white tan Skirtblouse 3 4 12 outcomes in sample space B Y B W B T B S Y Y W Y T Y S R Y R W R T R S Counting Principle 2 events: m · n 3 · 4 = 12

15. Course: Math Lit. Aim: Counting Principle Model Problem In horse racing, a trifecta consists of choosing the exact order of the first three horses across the finish line. If there are eight horses in a race, how many trifectas are possible, assuming no ties. 1 st place2 nd place3 rd place 876 x x = 336 Nine runners are entered in a 100-meter dash for which a gold, silver, and bronze medal will be awarded for 1 st, 2 nd and 3 rd place finishes. In how many ways can the medals be awarded? 504

16. Course: Math Lit. Aim: Counting Principle Counting with and without Replacement From the letters a, b, c, d, and e, how many four letter groups can be formed if a.a letter can be used more than once? b.each letter can be used exactly once? 1 st 2 nd 3 rd 4 th 5555... = 5 4 = 625 1 st 2 nd 3 rd 4 th 5432... = 120

17. Course: Math Lit. Aim: Counting Principle Model Problem A four-digit serial number is to be created from the digits 0 through 9. How many of these serial numbers can be created if 0 can not be the first digit, no digit may be repeated, and the last digit must be 5? 1) 448 2) 2240 3) 504 4) 2,520 possible outcomes E1E2 E3 E4 8 87 1    = 448 0, 1, 2, 3, 4, 5, 6, 7, 8, 910 possible outcomes to start

18. Course: Math Lit. Aim: Counting Principle Determine the number of outcomes: 4 coins are tossed A die is rolled and a coin is tossed A tennis club has 15 members: 8 women and seven men. How many different teams may be formed consisting of one woman and one man on each team? A state issues license plates consisting of letters and numbers. There are 26 letters, and the letters may be repeated on a plate; there are 10 digits, and the digits may be repeated. The how many possible license plates the state may issue when a license consists of: 2 letters, followed by 3 numbers, 2 numbers followed by 3 letters.