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Name11-1ClassDateAdditional Vocabulary SupportPermutations and CombinationsFundamental Counting PrincipleIf event M can occur in m ways and is followed by event N that can occur in n ways, thenevent M followed by event N can occur in m ? n ways.Example 4 different fruits and 6 different vegetables give 4 ? 6 possible fruit andvegetable combinations.Solve.1. Hector has 6 computers and 7 printers to choose from. How many possiblecomputer-printer combinations can he make?422. Raymond and Jasmine have 8 sofas and 14 chairs to choose between. Howmany possible sofa-chair combinations can they make?112Number of PermutationsThe number of permutations of n items of a set arranged r items at a time isn!n Pr 5 (n 2 r)!for 0 # r # nExample8 P358!8!5 5! 5 336(8 2 3)!Evaluate each expression.3. 7 P4 58404. 9 P5 515,120Number of CombinationsThe number of combinations of n items of a set chosen r items at a time isnC r5n!for 0 # r # nr!(n 2 r)!Example7C 357!7!7!5 3! ? 4! 5 6 ? 24 5 353!(7 2 3)!Evaluate each expression.5. 8 C5 5566. 9 C3 584Prentice Hall Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.1

Name11-1ClassDateThink About a PlanPermutations and CombinationsConsumer Issues A consumer magazine rates televisions by identifying two levelsof price, five levels of repair frequency, three levels of features, and two levels ofpicture quality. How many different ratings are possible?Understanding the Problem1. How many levels of price are possible? 22. How many levels of repair frequency are possible? 53. How many levels of features are possible? 34. How many levels of picture quality are possible? 25. What is the problem asking you to determine?the number of different ratings that are possiblePlanning the Solution6. What is the Fundamental Counting Principle?If an event M can occur in m ways and is followed by event N that can occur in nways, then event M followed by event N can occur in m n ways.7. How can the Fundamental Counting Principle help you solve the problem?Answers may vary. Sample: Each rating type is an event, and I know thepossible number of ways each event can occur. So by the FundamentalCounting Principle, the total number of ratings is the product of thenumber of ways each event can occur.Getting an Answer8. Write an expression for the number of different ratings that are possible.9. How many different ratings are possible? 60Prentice Hall Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.2z2?5?3?2z

Name11-1ClassDatePracticeForm GPermutations and Combinations1. How many 2-letter pairs of 1 vowel and 1 consonant can you make from theEnglish alphabet? Consider “y” to be a consonant. 1052. An ice cream shop offers 33 flavors of ice cream and 7 toppings. How manydifferent sundaes can the shop make using 1 flavor and 1 topping? 2313. A contest winner gets to choose 1 of 8 possible vacations and bring 1 of 10friends with her. How many different ways could the contest winner selecther prize? 80Evaluate each expression.4. 8! 40,32011!5. 9! 1109!8. 2!6! 2529. 3(7!) 15,1206. 6!4! 17,2807. 3(5!) 36010!10. 5! 30,2403!8!11. 5! 201612. An art gallery plans to display 7 sculptures in a single row.a. How many different arrangements of the sculptures are possible? 5040b. If one sculpture is taken out of the show, how many different arrangementsare possible? 720Evaluate each expression.13. 12P11 479,001,600 14. 12P10 239,500,800 15. 12P5 95,04016. 12P1 1217. 5P2 2020. 6P2 3018. 7P4 84019. 8P6 20,16021. In how many ways can four distinct positions for a relay race be assigned froma team of nine runners? 3024Evaluate each expression.22. 12C11 1223. 12C10 6624. 12C5 79226. 12C12 127. 5C4 1 5C3 1528. C 15 25C325. 12C1 1229. 4 (7C2) 8430. Thirty people apply for 10 job openings as welders. How many differentgroups of people can be hired? 30,045,015Prentice Hall Gold Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.3

NameClass11-1DatePractice (continued)Form GPermutations and CombinationsFor each situation, determine whether to use a permutation or a combination.Then solve the problem.31. You draw the names of 5 raffle winners from a basket of 50 names. Eachperson wins the same prize. How many different groups of winners could youdraw? combination: 2,118,76032. A paint store offers 15 different shades of blue. How many different ways couldyou purchase 3 shades of blue? combination: 45533. How many different 5-letter codes can you make from the letters in the wordcipher? permutation: 720Assume a and b are positive integers. Determine whether each statement is trueor false. If it is true, explain why. If it is false, give a counterexample.34. a!b! 5 b!a! True; CommutativeProperty of Multiplication35. (a2)! 5 (a!)2 False; leta 5 2: (22)! 5 24; (2!)2 5 436. a ? b! 5 (ab)! False; let a 5 2 andb 5 3: 2 ? 3! 5 12; (2 ? 3)! 5 72037. (a 1 0)! 5 a!True; Identity Property of Addition38.a!a5 Q b R ! False; let a 5 4 andb!39. a!(b! 1 c!) 5 a!b! 1 a!c!True; Distributive Property5 245 12; Q 4R ! 5 2! 5 2b 5 2: 4!2!2240. A restaurant offers a fixed-priced meal of 1 appetizer, 1 entrée, 2 sides, and1 dessert. How many different meals could you choose from 4 appetizers,5 entrees, 8 sides, and 3 desserts? 168041. Writing Explain the difference between a permutation and a combination. Answersmay vary. Sample: If you choose r items from a group of n items and the order in whichthe items are chosen is important, then it is a permutation. If the order does not matter,then it is a combination.42. Reasoning Show that for n 5 r, the value of nCr 5 1.nCnn!n!5 n!(n n!2 n)! 5 n!0!5 n!(1)51Prentice Hall Gold Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.4

Name11-1ClassDatePracticeForm KPermutations and CombinationsUse the Fundamental Counting Principle to solve the following problems.1. You must make a password for your email account. The password must consistof two letters followed by four digits. How many different passwords arepossible? 6,760,000 passwords2. Your father is buying a sport coat, a pair of pants, and a tie. Sport coats comein 6 different colors. Pants come in 4 different colors. There are 25 different tiestyles to choose from. How many different combinations are possible?600 combinationsEvaluate each expression.3. 6! 7209!5. 7! 724. 5!4! 2880Find the number of permutations in the following problems.6. Your coach has twelve team jerseys numbered from 1 through 12. He plans togive one jersey to each of the twelve members of the basketball team. In howmany ways can the jerseys be assigned?479,001,600 ways7. The owner of a car lot is lining up 7 cars in the show-room window. In howmany ways can the cars be ordered? 5040 waysEvaluate each expression.8. 5 P3 6010. 11P5 55,4409. 8 P5 672011. Twelve different types of pizza are being judged in a contest. In how manydifferent ways can the pizzas be judged first, second, third, and fourth?11,880 waysPrentice Hall Foundations Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.5

Name11-1ClassDatePractice (continued)Form KPermutations and CombinationsEvaluate each expression.12. 7C2 2113. 9C5 12614. 12C7 79215. 8C6 2816. 5(6C3) 10017. 10C7 1 5C2 130Decide whether to use a permutation or a combination for each situation. Thensolve the problem.18. An ice cream parlor offers 14 different types of ice cream. In how manydifferent ways can you select 5 types of ice cream to sample?combination; 2002 ways19. Eleven groups entered a science fair competition. In how many ways can thegroups finish first, second, and third?permutation; 990 ways20. Your aunt is ordering appetizers for her and her family. The restaurant offers10 different appetizers. She will select 4 appetizers. How many differentcombinations of appetizers can your aunt possibly select?combination; 210 combinations21. Error Analysis Your friend is shopping for blue jeans. The clothing store offers18 different types of blue jeans, and your friend will buy 5 different types.Your friend believes that she has 1,028,160 different combinations that shecould possibly select. What error did your friend make? How many differentcombinations could she possibly select?Your friend used a permutation when she should have used a combination.She could possibly select 8568 different combinations.Prentice Hall Foundations Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.6

NameClassDateStandardized Test Prep11-1Permutations and CombinationsMultiple ChoiceFor Exercises 1–5, choose the correct letter.1. You choose 5 apples from a case of 24 apples. Which best represents thenumber of ways you can make your selection? B5C1924C55P2419P52108402. Which is equivalent to 7P3? H28353. A traveler can choose from three airlines, five hotels, and four rental car companies.How many arrangements of these services are possible? B1260220495ba!b!(a!)15,12045,0004. Which is equivalent to a!(b!)? I(ab)!(ab!)!5. Which is equivalent to 9C5? A1263024Short Response6. You have a 1 bill, a 5 bill, a 10 bill, a 20 bill, a quarter, a dime, a nickel, anda penny. How many different total amounts can you make by choosing 6 billsand coins? Show your work.[2] 8C6 558!6!(8 2 6)!8 ? 7 ? 6 ? 5 ? 4 ? 3 ? 2 ? 1(6 ? 5 ? 4 ? 3 ? 2 ? 1)(2 ? 1)5 5625 28[1] incorrect or incomplete work shown[0] incorrect answer and no work shown OR no answer givenPrentice Hall Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.7

Name11-1ClassDateEnrichmentPermutations and CombinationsDinner at a Chinese RestaurantA typical Chinese restaurant will often feature a Special Dinner, in which thecustomer has the choice of ordering one appetizer and one entree.1. If there are 8 appetizers and 11 entrees, how many different Special Dinnersare there? 882. If there are 12 appetizers and 7 entrees, how many different Special Dinnersare there? 843. If there are A appetizers and E entrees, how many different Special Dinnersare there? AE4. There are 12 appetizers; 4 are soups; 6 contain meat, and 2 do not. In howmany different orders can 3 different appetizers be brought to the table? 13205. In how many different orders can 5 different appetizers of the 12 be broughtto the table? 95,0406. Do Exercises 1–5 involve permutations or combinations? permutations7. Assume that 3 customers arrive and order different appetizers to share from achoice of 12 appetizers.a. Does this problem involve permutations or combinations? combinationsb. Why? order doesn’t matterc. In how many possible ways can this be done? 12C3 5 2208. Suppose that 5 customers arrive, and each orders a different appetizer toshare from a choice of 12 appetizers. In how many ways can this be done?12C59. Suppose that 7 customers arrive, and each orders a different appetizer toshare from a choice of 12 appetizers.a. In how many ways can this be done? 12C7 5 792b. Why is this answer the same as the number of ways that 5 customers canorder different appetizers? nCr 5 nCn2r or 12C5 5 12C7Prentice Hall Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.85 792

NameClass11-1DateReteachingPermutations and CombinationsIf you select some items out of a group and the order of the items in your selectionis important, then your selection is a permutation of the group.For example, suppose Ana, Bob, Cal, and Dan enter a local essay contest. Here are somepossible ways for the judges to select the first-prize and second-prize winners.“Ana, Bob” means Ana is first and Bob is second.First PrizeSecond PrizeAnaBob“Bob, Ana” means Bob is first and Ana is second.DanCalBobAnaBobDanThe order of the names in the selection isimportant. The selection “Ana, Bob” is apermutation of the group of contestants.The number of permutations of n items of a set arranged r items at a time isn Pr5n! ,(0 # r # n)(n 2 r)!ProblemIn how many ways can the judges select the first-prize and second-prize winnersin the essay contest described above?Step 1Is the order of the names in each selection important?Yes. “Ana, Bob” is not the same as “Bob, Ana.” You are looking for the total numberof permutations of 2 items each selected from a group of 4 items.Step 2Describe n and r.There are 4 people in the group of contestants. n 5 4There are 2 people in each selection of prize winners. r 5 2Step 3Substitute for each variable in the formula.n Pr5 4P2 54?3?2?1n!4!4!55 2! 55 122?1(n 2 r)!(4 2 2)!There are 12 ways for the judges to choose the first-prize and second-prize winners.Exercises1. In how many ways can you choose 6 letters for a password from the set A, B, E,L, N, O, S, T, Y? 60,4802. In how many ways can a club with 15 members elect a president, vicepresident, secretary, and treasurer? 32,7603. In how many ways can a family of 6 line up in 1 row for a photograph? 720Prentice Hall Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.9

NameClass11-1DateReteaching (continued)Permutations and CombinationsIf you select some items out of a group and the order of the items in your selectionis NOT important, then your selection is a combination of the group.For example, suppose Ana, Bob, Cal, and Dan enter a local essay contest. Twofinalists will go to the statewide essay contest. Here are some ways for the judgesto select the contestants who will go to the state contest.“Ana, Bob” means both Ana and Bob will go.FinalistFinalistAnaBob“Bob, Ana” means both Ana and Bob will go.DanCalBobAnaBobDanThe order of the names in the selection isNOT important. The selection “Ana, Bob” is acombination of the group of contestants.The number of combinations of n items of a set chosen r items at a time isnCr5n!r!(n 2 r)!, (0 # r # n)ProblemIn how many ways can the judges select the finalists who will go to the statecontest?Step 1 Is the order of the names in each selection important?No. “Ana, Bob” has the same meaning as “Bob, Ana.” You are looking for the totalnumber of combinations of 2 items each selected from a group of 4 items.Step 2 Describe n and r.There are 4 people in the group of contestants. n 5 4There are 2 people in each selection of contestants going on to state. r 5 2Step 3 Substitute for each variable in the formula.n!4!4!4?3?2?15 4C2 55555 122 5 6r!(n 2 r)!2!(4 2 2)!2!(2!)2 ? 1(2 ? 1)There are 6 ways for the judges to choose the finalists going to the state contest.nCrExercises4. You have 12 CDs, but only have time to listen to 2 of them. How manycombinations of CDs do you have to choose from? 665. Your biology teacher chooses 6 students from a class of 26 to do a specialproject. How many different groups can your teacher form? 230,2306. How many 3-flavor blends can you create from 10 frozen yogurt flavors? 120Prentice Hall Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.10

Name11-2ClassDateAdditional Vocabulary SupportProbabilityComplete the vocabulary chart by filling in the missing information.Word . the number of times the eventoccurs divided by the numberof trialsYou take 12 marbles from a bagand 3 of them are blue. Theexperimental probability ofpulling a blue marble from the3bag is 12 5 0.25 5 25%.simulation2. a simulation is a model of aneventYou can simulate guessing on aset of true or false questions byflipping a coin.sample spacea list of all possible outcomes to anexperiment or activity3. The sample space for a flipof a coin is heads or tails.equally likelysample space4. a sample space in which eachoutcome has the same chanceof occurringThe sample space for randomlyselecting a card from a deck of 52cards includes all of the cards inthe deck, and each outcome hasan equal chance of occurring.theoreticalprobabilityIf an event A occurs in m out of nequally likely outcomes, then the5. The theoretical probabilityof pulling an Ace from amtheoretical probability of A is n .41deck of 52 cards is 525 13.Prentice Hall Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.11

NameClassDateThink About a Plan11-2ProbabilityLottery A lottery has 53 numbers from which five are drawn at random. Eachnumber can only be drawn once. What is the probability of your lottery ticketmatching all five numbers in any order?Know1. The lottery hasz53z possible numbers that can be drawn.2. Each number can be drawn3. A total ofz5z1z time(s).z numbers will be drawn.Need4. To solve the problem I need to find:the theoretical probability of the numbers on a lottery ticket matching thenumbers drawn in any order.Plan5. Because order does not matter, the size of the sample space is azcombination6. What is the sample space?all combinations of 53 numbers chosen 5 at a time7. What is the size of the sample space? 53C5 5 2,869,6858. How many of the events in the sample space represent your ticket? 19. What is the probability of your lottery ticket matching all five numbers in anyorder?1N 0.000000352,869,685Prentice Hall Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.12z.

Name11-2ClassDatePracticeForm GProbability1. A basketball player attempted 24 shots and made 13. Find the experimentalprobability that the player will make the next shot she attempts. N 0.54 or 54%2. A baseball player attempted to steal a base 70 times and was successful 47times. Find the experimental probability that the player will be successful onhis next attempt to steal a base. N 0.67, or 67%Graphing Calculator For Exercises 3–4, define a simulation by telling how yourepresent correct answers, incorrect answers, and the quiz. Use your simulationto find each experimental probability.3. If you guess the answers at random, what is the probability of getting at leastthree correct answers on a four-question true-or-false quiz?Answers may vary. Sample: Let “1” be a correct answer. Let “2” be an incorrect answer.5Generate 16 sets of 4 random 1’s and 2’s; 165 0.3125 N 0.31, or 31%.4. A five-question multiple-choice quiz has four choices for each answer. If youguess the answers at random, what is the probability of getting at least fourcorrect answers? Answers may vary. Sample: Let “1” be a correct answer. Let “2”,“3”, and “4” be incorrect answers. Generate 64 sets of 5 random 1’s, 2’s, 3’s, and 4’s;164 5 0.015625 N 0.02, or 2%.A group of five cards are numbered 1–5. You choose one card at random. Findeach theoretical probability.5. P(card is a 2) 15 5 0.20, or 20%6. P(even number) 25 5 0.40, or 40%7. P(prime number) 35 5 0.60, or 60%8. P(less than 5) 45 5 0.80, or 80%A bucket contains 15 blue pens, 35 black pens, and 40 red pens. You pick onepen at random. Find each theoretical probability.5510. P(blue pen or red pen) 90 N 0.61, 61%9. P(black pen) 3590 N 0.39, 39%11. P(not a blue pen) 7590 N 0.83, 83%12. P(black pen or not a red pen)5090N 0.56, 56%Prentice Hall Gold Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.13

Name11-2ClassDatePractice (continued)Form GProbability13. There are 225 juniors and 255 seniors at your school. The school chooses 5juniors and seniors as Student All-Stars. What is the theoretical probabilitythat exactly 2 of the Student All-Stars will be juniors? N 0.33, or 33%The rectangular yard shown below has a circular pool and a triangular garden.A ball from an adjacent golf course lands at a random point within the yard.Find each theoretical probability.14. The ball lands in the pool.N 0.09, or 9%8 ft6 ft15. The ball lands in the gardenN 0.005, or 0.5%50 ft12 ft16. The ball lands in the garden or the pool.N 0.095, or 9.5

Prentice Hall Gold Algebra 2 Teaching Resources . Answers Property of Multiplication b 5 3: 2? 3! 5 12; (2? 3)! 5 720 b 5 2: 4! 2! 5 24 2 5 12; Q4 2 R! 5 2! 5 2 a 5 2: (2 2)! 5 24; (2!) 5 4 True; Identity Property of Addition True; Distributive Property True; Commutative False; let a 5 2 and

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