MATH 30-2



MATH 30-2PERMUTATIONS, COMBINATIONS, & THE FUNDAMENTAL COUNTING PRINCIPLE - Module THREEModule 3 - Assignment Booklet Student: _____________________________ Date Submitted: ______________________ Lesson 1: Fundamental Counting Principle1.In Canada, all postal codes are made up of six characters, which follow the pattern letter-digit-letter-digit-letter-digit. The first letter of the code identifies a specific region. Each of the remaining two letters in the postal code can be any of 20 letters, and each of the three digits can be any digit from 0 to 9. All letters and digits can be repeated.In Alberta, all postal codes start with the letter T. How many different postal codes are possible in Alberta?2.Each switch on a panel of 5 light switches may be in an “on” or “off” position at any given time. One possible unique position of all 5 switches is shown.Find the number of possible unique positions for all 5 light switches.3.In 1912, Alberta licence plates consisted of four digits. Each digit could be repeated, but the first digit could not be zero. By 1941, Alberta licence plates consisted of five digits. Each digit could be repeated, but the first digit could not be pared with the number of licence plates available in 1912, find the increase in the number of licence plates available in 1941.4.Recall the example about Louie’s Bistro. You had a choice of 2 salads, mixed or Caesar. You had a choice of 4 entrées: chili, club sandwich, roasted chicken, or grilled fish. Finally, for dessert, you could choose a brownie, a cupcake, or a piece of cheesecake. How many different meals are possible if you do not have to choose an item from each category? Explain your solution.5. Karen has a bank card that requires a 4-digit PIN number. All digits from 0 to 9 can be used.a.How many PINs are possible if the digits can repeat?b.How many PINs are possible if the digits cannot repeat?6. A group of 8 teenagers are lined up to buy tickets to a rock concert. How many possible arrangements of teenagers are there? Leave your answer in factorial notation.Lesson 2: Factorial Notation1.Simplify .2.Solve the equation for n.3.Nine runners have made it to the zone cross-country running finals. Hariette is considered the favourite to win based on her previous times. If she does win, in how many different orders can the other racers finish?4.A local hockey club is setting up an online system for registration for its players. Seven-character passwords will be assigned to each player—each password will start with the letter S to indicate the home club’s team name, Sharks. The other digits in the password are made up of the numbers 4 through 9, and no digit can be repeated. How many passwords can the hockey club create?Lesson 3: Problems Involving Permutations1.Manny needs to create a 4-digit password for his cell phone. If no digits can be repeated, in how many ways can this be done?2.How many 3-letter arrangements can be made from the letters in VERTICAL if no letter can be used more than once and each arrangement is made up of a vowel between two consonants?3.In a gymnastics club, 10 gymnasts practise forming a human pyramid. The pyramid uses only 6 gymnasts at any one time. There are 3 gymnasts in the bottom row, 2 in the second row, and 1 at the top.Hemera/ThinkstockIf any gymnast can fill any position in the pyramid, how many different arrangements of gymnasts are there?4.Bart wants to plant 8 trees in a row along his fence. He has been given 4 birches, 1 spruce, 1 poplar, 1 willow, and 1 elm. If the 4 birches are identical, then how many possible arrangements of trees are there?5.Solve for r.5Pr = 60Lesson 4: Solving Problems That Involve Combinations1.A car dealership is promoting a particular model of car that has 8 different optional features available. Each optional feature can be purchased separately.a.How many different packages of 3 optional features are possible for this model of the car?b.Explain why the number of different packages of 5 optional features is the same as the number of different packages of 3 optional features for this model of car.c.Another model of car has n different optional features available. When 2 optional features are chosen for this model of car, there are 45 different packages available. Determine algebraically the number of optional features, n, that are available for this model of car.2.a.How many diagonals are there in an octagon?b.Why is 2.a. an example of a combination problem and not a permutation problem?3.How many 6-member committees can be formed from 8 girls and 9 boys if a.there are no conditionsb.there must be exactly 3 girlsc.there must be at least 4 boysLesson 5: Solving Probability Problems That Involve Permutations and Combinations1.A person has bought one ticket each for two separate lotteries. Determine the probability of winning a grand prize for the two lotteries.a.For the first lottery, 5 random numbers are drawn from 40. To win the grand prize, you need to select the correct 5 numbers.b.For the second lottery, 4 random numbers are drawn one at a time from 35. To win the grand prize, you need to select the correct 4 numbers in the correct order.c.A person is more likely to win the first lottery described in question 1.a., even though the first lottery requires a person to have more correct numbers (5 compared to 4) selected from a larger set of numbers (40 compared to 35). Explain why this is true.2.A zookeeper is setting up an African exhibit and has found the following animals available from another zoo.AntelopeNumber AvailableLarge CatsNumber AvailableSmall AnimalsNumber Availablecommon eland1African leopard2Abyssinian genet4greater kudu1Cape lion1common genet2nyala2Northwest African cheetah2Cape ratel6red hartebeest2civet1black-backed jackal2side-striped jackal3a.If the zookeeper randomly selects an animal, what is the probability that a red hartebeest will be chosen?b.If the zookeeper randomly selects 8 animals, what is the probability that a cheetah or a jackal will not be chosen?c.If the zookeeper randomly selects 4 small animals, what is the probability that at least 1 Abyssinian genet will be chosen?d.If the zookeeper asked for 4 antelope or large cats, what is the probability that both the eland and the kudu would be received?3.a.Suppose you wrote the following message using the pigpen cipher from the beginning of the lesson.What is the probability that an interceptor would correctly guess the message on the first try if the interceptor assumed each symbol corresponded to a letter of the alphabet?b.What is the probability of the person correctly guessing the following message on the first try if that person knew was an s? (Hint: You can assume that if you guess the first correctly, you will also guess the second correctly.)MODULE 3 SUMMARYIn this module you learned how to:Determine the number of ways in which events can occur. You investigated the use of permutations and combinations in determining such things as the possible match-up of teams for the playoffs, the number of codes that can be used to open a garage, and the probability of winning a favourite game.How counting methods are used to solve problems involving codes and passwords.Some of the key ideas that you learned in each lesson are:how to apply the fundamental counting principle. If one task can be done in a ways and a second in b ways, then the two tasks may be completed in a ? b ways. how to apply and simplify factorial notation.Factorial notation represents the product of consecutive natural numbers.n! = n(n ? 1)(n ? 2)(n ? 3)…(3)(2)(1)how to use this knowledge to solve equations involving the use of permutation and combination formulas.how to apply factorial notation to problems such as determining the starting line-up for a baseball team.that permutations are an arrangement of items when order is important.you applied your knowledge of permutations to problems where order mattered, such as a horse race.that combinations are a selection of items where order is not important.using combinations, you determined the number of chords that could be played on a piano.you investigated probability problems involving counting methods. you analyzed problems to determine which counting method would be most appropriate in determining the probability of an event happening.In Module 4 you will explore how some problems can be modelled and solved using polynomialsMODULE 3 – PERMUTATIONS, COMBINATIONS, & THE FUNDAMENTAL COUNTING PRINCIPLE SUMMATIVE ASSIGNMENTComplete the following questions from your text book. Show steps completely and clearly, as marks are assigned for mathematical literacy and communication. Always use graph paper, rulers, and pencils as necessary. Attach questions and study notes securely to this booklet before you hand everything in.Text: Principles of Mathematics 12 - Chapter 2: Counting MethodsSection 2.1: Pages: 73 to 75 # 5, 8, 14Section 2.2: Pages: 81 to 83 # 3all, 5e,f, 6a,b,e, 11a,c, 15Section 2.3: Pages: 93 to 95 # 1d,f, 3a, 5, 8, 14, 15Section 2.4: Pages: 104 to 107 # 6a, 9a, 11a, 13, Section 2.5: Pages: 110 #2Section 2.6: Pages: 118 to 120 # 4a,c,d, 8, 11a,b, 14aiii, 15a,c,18Section 2.7: Pages: 126 to 128 # 3, 4, 5Module 3 is now complete. Once you have received your corrected work, review your instructor’s comments and prepare for your module three test. ................
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