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Activating Activity • If ice cream sundaes come in 5 flavors with 4 possible toppings, how many different sundaes can be made with one flavor of ice cream and one topping? 1 Counting Principle Permutations Combinations How do you use the counting principle, permutations and combinations to predict outcomes of different events or occurrences? 2 The Fundamental Counting Principle: If there are a ways for one activity to occur, and b ways for a second activity to occur, then there are a • b ways for both to occur. 3 Examples: 1. Activities: roll a die and flip a coin There are 6 ways to roll a die and 2 ways to flip a coin. There are 6 • 2 = 12 ways to roll a die and flip a coin. 2. Activities: draw two cards from a standard deck of 52 cards without replacing the cards There are 52 ways to draw the first card. There are 51 ways to draw the second card. There are 52 • 51 = 2,652 ways to draw the two cards. 4 The Counting Principle also works for more than two activities. 3. Activities: a coin is tossed five times There are 2 ways to flip each coin. There are 2 • 2 • 2 • 2 •2 = 32 arrangements of heads and tails. 4. Activities: a die is rolled four times There are 6 ways to roll each die. There are 6 • 6 • 6 • 6 = 1,296 possible outcomes. Remember: The Counting Principle is easy! Simply MULTIPLY the number of ways each activity can occur. 5 A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? Information about girls' ice skates: Colors: white, beige, pink, yellow, blue Sizes: 4, 5, 6, 7, 8 Extras: tassels, striped laces, bells Assuming that all skates are sold with ONE extra, how many possible arrangements exist? 6 Your state issues license plates consisting of letters and numbers. There are 26 letters and the letters may be repeated. There are 10 digits and the digits may be repeated. How many possible license plates can be issued with two letters followed by three numbers? The ice cream shop offers 31 flavors. You order a double-scoop cone. In how many different ways can the clerk put the ice cream on the cone if you wanted two different flavors? 7 The local Family Restaurant has a daily breakfast special in which the customer may choose one item from each of the following groups: Breakfast Accompan Juice Sandwich iments egg and breakfast orange ham potatoes cranberry egg and apple slices tomato bacon fresh fruit apple egg and cup grape cheese pastry a.) How many different breakfast specials are possible? b.) How many different breakfast specials without meat are possible? 8 Permutation: A set of objects in which position (or order) is important. To a permutation, the trio of Brittany, Alan and Greg is DIFFERENT from Greg, Brittany and Alan. Permutations are persnickety (picky). Combination: A set of objects in which position (or order) is NOT important. To a combination, the trio of Brittany, Alan and Greg is THE SAME AS Greg, Brittany and Alan. 9 Let's look at which is which: Permutation versus Combination 1. Picking a team captain, 1. Picking three team members pitcher, and shortstop from a from a group. group. 2. Picking your favorite two 2. Picking two colors from a colors, in order, from a color color brochure. brochure. 3. Picking first, second and 3. Picking three winners. third place winners. 10 Formulas: A permutation is the choice of r things from a set of n things without replacement and where the order matters. Remember: n is the total number of objects r is the number of objects chosen (want) 11 1. Compute: 5 P5 2. Compute: 6P2 3. Find the number of ways to arrange 5 objects that are chosen from a set of 7 different objects. 4. What is the total number of possible 5-letter arrangements of the letters w, h, i, t, e, if each letter is used only once in each arrangement? 5. How many different 3-digit numerals can be made from the digits 4, 5, 6, 7, 8 if a digit can appear just once in a numeral? 12 1. Compute: 5 P5 5 · 4 · 3 · 2 · 1 = 120 2. Compute: 6P2 6 · 5 = 30 or multiply by two factors of the factorial, starting with 6 3. Find the number of ways to arrange 5 objects that are chosen from a set of 7 different objects. 7P5 = 7·6·5·4·3 = 2520 or 4. What is the total number of possible 5-letter arrangements of the letters w, h, i, t, e, if each letter is used only once in each arrangement? 5 P5 = 5·4·3·2·1 = 120 or or simply 5! 5. How many different 3-digit numerals can be made from the digits 4, 5, 6, 7, 8 if a digit can appear just once in a numeral? 5 P3 = 5·4·3 = 60 or 13 A permutation is an arrangement of objects in specific order. The order of the arrangement is important!! Consider, four students walking toward their school entrance. How many different ways could they arrange themselves in this side-by-side pattern? 1,2,3,4 2,1,3,4 3,2,1,4 4,2,3,1 1,2,4,3 2,1,4,3 3,2,4,1 4,2,1,3 1,3,2,4 2,3,1,4 3,1,2,4 4,3,2,1 1,3,4,2 2,3,4,1 3,1,4,2 4,3,1,2 1,4,2,3 2,4,1,3 3,4,2,1 4,1,2,3 1,4,3,2 2,4,3,1 3,4,1,2 4,1,3,2 The number of different arrangements is 24 or 4! = 4 • 3 • 2 • 1. There are 24 different arrangements, or permutations, of the four students walking side-by-side. 14 In how many ways can 3 different vases be arranged on a tray? 15 A combination is the choice of r things from a set of n things without replacement and where order does not matter. 16 Example 1: Evaluate : Notice how the cancellation occurs, leaving only 2 of the factorial terms in the numerator. A pattern is emerging ... when finding a combination such as the one seen in this problem, the second value (2) will tell you how many of the factorial terms to use in the numerator, and the denominator will simply be the factorial of the second value (2). 17 Heather has finally narrowed her clothing choices for the big party down to 3 skirts, 2 tops and 4 pair of shoes. How many different outfits could she form from these choices? 18 A coach must choose five starters from a team of 12 players. How many different ways can the coach choose the starters? 19 Joleen is on a shopping spree. She buys six tops, three shorts and 4 pairs of sandals. How many different outfits consisting of a top, shorts and sandals can she create from her new purchases? (6)(3)(4) = 72 possible outfits What is the total number of possible 4-letter arrangements of the letters m, a, t, h, if each letter is used only once in each arrangement? or or simply 4! 20 3-2-1 Ticket out the door • Name three different terms used to predict outcomes of events. • Name two formulas used today. • Name one way to tell which concept you will choose to use to predict outcomes of events. 21