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1. x - 5 = -17 2. -3x + 12= -15 3. 20 – 3x = 17 4. = -10

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Probability The Counting Principle & Permutations

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Use an appropriate method to find the number of outcomes in each of the following situations: 1. Your school cafeteria offers chicken or tuna sandwiches; chips or fruit; and milk, apple juice, or orange juice. If you purchase one sandwich, one side item and one drink, how many different lunches can you choose? Sandwich(2)Side Item(2) Drink(3) Outcomes chicken tuna There are 12 possible lunches. chips fruit chips fruit apple juice orange juice milk chicken, chips, apple chicken, chips, orange chicken, chips, milk chicken, fruit, apple chicken, fruit, orange chicken, fruit, milk tuna, chips, apple tuna, chips, orange tuna, chips, milk tuna, fruit, apple tuna, fruit, orange tuna, fruit, milk

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Easier Way No need to make the tree diagram. Multiply each of the number of choices (2 sandwiches, 2 sides, 3 drinks)

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Counting Principle At a sporting goods store, skateboards are available in 8 different deck designs. Each deck design is available with 4 different wheel assemblies. How many skateboard choices does the store offer? 32

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Counting Principle A father takes his son, Marcus, to Wendys for lunch. He tells Marcus he can get a 5 piece nuggets, a spicy chicken sandwich, or a single for the main entrée. For sides, he can get fries, a side salad, potato, or chili. And for drinks, he can get a frosty, coke, sprite, or an orange drink. How many options for meals does Marcus have? 48

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Many iPods can vary the order in which songs are played. Your mp3 currently only contains 8 songs. Find the number of orders in which the songs can be played. 1st Song 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th Outcomes There are 40,320 possible song orders. In this situation it makes more sense to use the Fundamental Counting Principle. 8 The solutions in examples 3 and 4 involve the product of all the integers from n to one. The product of all positive integers less than or equal to a number is a factorial. 7 6 5 4 3 2 1= 40,320

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Factorial EXAMPLE with Songs eight factorial The product of counting numbers beginning at n and counting backward to 1 is written n! and its called n factorial. 8! = 8 7 6 5 4 3 2 1 = 40,320

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Factorial Simplify each expression. a.4! b.6! c. For the 8th grade field events there are five teams: Red, Orange, Blue, Green, and Yellow. Each team chooses a runner for lanes one through 5. Find the number of ways to arrange the runners. 4 3 2 1 = 24 6 5 4 3 2 1 = 720 = 5! = 5 4 3 2 1 = 120

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The student council of 15 members must choose a president, a vice president, a secretary, and a treasurer. President (15) Vice (14) Secretary (13) Treasurer (12) Outcomes There are 32,760 permutations for choosing the class officers. In this situation it makes more sense to use the Fundamental Counting Principle. 151413 12 = 32,760

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Lets say the student council members names were: John, Miranda, Michael, Kim, Pam, Jane, George, Michelle, Sandra, Lisa, Patrick, Randy, Nicole, Jennifer, and Paul. If Michael, Kim, Jane, and George are elected, would the order in which they are chosen matter? President Vice President Secretary Treasurer Is MichaelKim Jane George the same as… Jane Michael George Kim ?

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Permutation Notation

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Permutations ORDER MATTERS! Placement Examples: assigned seats, winning a race or running a race, 1 st place, 2 nd place, etc Positions Examples: Pres., Vice Pres, Sec, Tres. Specific job/chore Examples: Hand out markers, pass out papers, etc

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Permutation When deciding who goes 1 st, 2 nd, etc., order is important. * Note if n = r then n P r = n ! A permutation is an arrangement or listing of objects in a specific order. The order of the arrangement is very important !! The notation for a permutation: n P r = n is the total number of objects r is the number of objects selected (wanted)

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Permutations Simplify each expression. a. 12 P 2 b. 10 P 4 c. At a school science fair, ribbons are given for first, second, third, and fourth place, There are 20 exhibits in the fair. How many different arrangements of four winning exhibits are possible? 12 11 = 132 10 9 8 7 = 5,040 = 20 P 4 = 20 19 18 17 = 116,280

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Permutation

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