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AlgebraAlgebra and Trigonometry (MindTap Course List)DISCUSS ■ DISCOVER: Why Is ( n r ) the Same as C ( n , r ) ? This exercise explains why the binomial coefficients ( n r ) that appear in the expression of ( x + y ) n are the same as C ( n , r ) , the number of ways of choosing r objects from n objects. First, note that expanding a binomial using only the Distributive Property gives ( x + y ) 2 = ( x + y ) ( x + y ) = ( x + y ) x + ( x + y ) y = x x + x y + y x + y y ( x + y ) 3 = ( x + y ) ( x x + x y + y x + y y ) = x x x + x x y + x y x + x y y + y x x + y x y + y y x + y y y (a) Expand ( x + y ) 5 using only the Distribution Property. (b) Write all the terms that represent x 2 y 3 . These are all the terms that contain two x ’s and three y ’s. (c) Note that the two x ’s appear in all possible positions. Conclude that the number of terms that represent x 2 y 3 is C ( 5 , 2 ) . (d) In general, explain why ( n r ) in the Binomial Theorem is the same as C ( n , r ) .BuyFind*arrow_forward*

4th Edition

James Stewart + 2 others

Publisher: Cengage Learning

ISBN: 9781305071742

Chapter 14.1, Problem 94E

Textbook Problem

**DISCUSS ■ DISCOVER: Why Is **
**the Same as **
**? **This exercise explains why the binomial coefficients

(a) Expand

(b) Write all the terms that represent

(c) Note that the two

(d) In general, explain why

Algebra and Trigonometry (MindTap Course List)

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Ch. 14.1 - CONCEPTS The Fundamental Counting Principle says...Ch. 14.1 - CONCEPTS The number of ways of arranging r objects...Ch. 14.1 - CONCEPTS The number of ways of choosing r objects...Ch. 14.1 - CONCEPTS True or False? aIn counting combinations,...Ch. 14.1 - SKILLS 5-16 Evaluating Permutations and...Ch. 14.1 - SKILLS 5-16 Evaluating Permutations and...Ch. 14.1 - SKILLS 5-16 Evaluating Permutations and...Ch. 14.1 - SKILLS 5-16 Evaluating Permutations and...Ch. 14.1 - SKILLS 5-16 Evaluating Permutations and...Ch. 14.1 - SKILLS 5-16 Evaluating Permutations and...

Ch. 14.1 - 5-6 Evaluating Permutations and Combinations...Ch. 14.1 - 5-6 Evaluating Permutations and Combinations...Ch. 14.1 - 5-6 Evaluating Permutations and Combinations...Ch. 14.1 - 5-6 Evaluating Permutations and Combinations...Ch. 14.1 - 5-6 Evaluating Permutations and Combinations...Ch. 14.1 - 5-6 Evaluating Permutations and Combinations...Ch. 14.1 - 17-36Fundamental Counting Principal These...Ch. 14.1 - 17-36 Fundamental Counting Principle These...Ch. 14.1 - 17-36 Fundamental Counting Principle These...Ch. 14.1 - Multiple-Choice TestA multiple-choice test has...Ch. 14.1 - Phone NumbersTelephone numbers consist of seven...Ch. 14.1 - Running a RaceIn how many different ways can a...Ch. 14.1 - Restaurant MealsA restaurant offers the items...Ch. 14.1 - Multiple RoutesTowns A, B, C, and D are located in...Ch. 14.1 - Flipping a CoinA coin is flipped five times, and...Ch. 14.1 - Rolling a Pair of Dice A red die and a white die...Ch. 14.1 - Rolling Three Dice A red die, a blue die, and a...Ch. 14.1 - Choosing Outfits A girl has five skirts, eight...Ch. 14.1 - License Plate Standard automobile license plates...Ch. 14.1 - ID Numbers A companys employee ID number system...Ch. 14.1 - APPLICATIONS 17-36 Fundamental Counting...Ch. 14.1 - APPLICATIONS 17-36 Fundamental Counting...Ch. 14.1 - APPLICATIONS 17-36 Fundamental Counting...Ch. 14.1 - APPLICATIONS 17-36 Fundamental Counting...Ch. 14.1 - APPLICATIONS 17-36 Fundamental Counting...Ch. 14.1 - APPLICATIONS 17-36 Fundamental Counting...Ch. 14.1 - APPLICATIONS 37-40 Counting Subsets These...Ch. 14.1 - APPLICATIONS 37-40 Counting Subsets These...Ch. 14.1 - APPLICATIONS 37-40 Counting Subsets These...Ch. 14.1 - APPLICATIONS 37-40 Counting Subsets These...Ch. 14.1 - 4152Counting PermutationsThese exercises involves...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 41-52 Counting Permutations These exercises...Ch. 14.1 - 53-56 Distinguishable Permutations These exercises...Ch. 14.1 - 53-56 Distinguishable Permutations These exercises...Ch. 14.1 - 53-56 Distinguishable Permutations These exercises...Ch. 14.1 - 53-56 Distinguishable Permutations These exercises...Ch. 14.1 - Distributions A man bought three vanilla ice-cream...Ch. 14.1 - Room Assignments When seven students take a trip,...Ch. 14.1 - Work Assignments Eight workers are cleaning a...Ch. 14.1 - Transporting Students A group of 30 students is...Ch. 14.1 - 61-74CombinationsThese exercises involve counting...Ch. 14.1 - 61-74CombinationsThese exercises involve counting...Ch. 14.1 - 61-74CombinationsThese exercises involve counting...Ch. 14.1 - 61-74CombinationsThese exercises involve counting...Ch. 14.1 - 61-74CombinationsThese exercises involve counting...Ch. 14.1 - 61-74CombinationsThese exercises involve counting...Ch. 14.1 - 61-74CombinationsThese exercises involve counting...Ch. 14.1 - APPLICATIONS 61-74 CombinationsThese exercises...Ch. 14.1 - APPLICATIONS 61-74 CombinationsThese exercises...Ch. 14.1 - APPLICATIONS 61-74 CombinationsThese exercises...Ch. 14.1 - APPLICATIONS 61-74 CombinationsThese exercises...Ch. 14.1 - APPLICATIONS 61-74 CombinationsThese exercises...Ch. 14.1 - APPLICATIONS 61-74 CombinationsThese exercises...Ch. 14.1 - APPLICATIONS 61-74 CombinationsThese exercises...Ch. 14.1 - APPLICATIONS 75-90 Counting PrinciplesSolve these...Ch. 14.1 - APPLICATIONS 75-90 Counting PrinciplesSolve these...Ch. 14.1 - APPLICATIONS 75-90 Counting PrinciplesSolve these...Ch. 14.1 - APPLICATIONS 75-90 Counting PrinciplesSolve these...Ch. 14.1 - Dance Committee A school dance committee is to...Ch. 14.1 - Casting a Play A group of 22 aspiring thespians...Ch. 14.1 - Hockey Lineup A hockey team has 20 players, of...Ch. 14.1 - Choosing a Pizza A pizza parlour offers four sizes...Ch. 14.1 - Choosing a Committee In how many ways can a...Ch. 14.1 - Parking Committee A five person committee...Ch. 14.1 - Arranging Books In how many ways can five...Ch. 14.1 - Arranging a Class PictureIn how many ways can ten...Ch. 14.1 - Seating ArrangementsIn how many ways can four men...Ch. 14.1 - Seating ArrangementsIn how many ways can four men...Ch. 14.1 - Selecting Prizewinners From a group of 30...Ch. 14.1 - Choosing a Delegation Three delegates are to be...Ch. 14.1 - DISCUSS: Pair of InitialsExplain why in any group...Ch. 14.1 - DISCUSS:Complementary CombinationsWithout...Ch. 14.1 - DISCUSS:An Identity Involving CombinationsKelvin...Ch. 14.1 - DISCUSS DISCOVER: Why Is (nr)the Same as...Ch. 14.2 - The set of all possible outcomes of an experiment...Ch. 14.2 - Let E and F be events in a sample space S. aThe...Ch. 14.2 - The conditional probability of E given that F...Ch. 14.2 - Let E and F be events in a sample space S. aThe...Ch. 14.2 - Rolling a Die An experiment consists of rolling a...Ch. 14.2 - Tossing a coin An experiment consists of tossing a...Ch. 14.2 - 720 Probability by Counting These exercises...Ch. 14.2 - 720 Probability by Counting These exercises...Ch. 14.2 - 9-10 A die is rolled. Find the probability of the...Ch. 14.2 - 9-10 A die is rolled. Find the probability of the...Ch. 14.2 - 1112 A card is drawn randomly from a standard...Ch. 14.2 - 1112 A card is drawn randomly from a standard...Ch. 14.2 - 1314 A ball is drawn randomly from a jar that...Ch. 14.2 - 1314 A ball is drawn randomly from a jar that...Ch. 14.2 - A poker hand, consisting of five cards, is dealt...Ch. 14.2 - Three CDs are picked at random from a collection...Ch. 14.2 - Two balls are picked at random from a jar that...Ch. 14.2 - A letter is chosen at random from the word...Ch. 14.2 - A five-card poker hand is drawn from a standard...Ch. 14.2 - A pair of dice is rolled, and the numbers showing...Ch. 14.2 - Union of EventsThese exercises involve finding the...Ch. 14.2 - Union of EventsThese exercises involve finding the...Ch. 14.2 - A die is rolled, and the number showing is...Ch. 14.2 - A die is rolled, and the number showing is...Ch. 14.2 - 25-26 A card is drawn at random from a standard 52...Ch. 14.2 - 25-26 A card is drawn at random from a standard 52...Ch. 14.2 - 27-32 Conditional ProbabilityThese exercises...Ch. 14.2 - 27-32 Conditional ProbabilityThese exercises...Ch. 14.2 - 29-30 Refer to the spinner in Exercises 2122. Find...Ch. 14.2 - 29-30 Refer to the spinner in Exercises 2122. Find...Ch. 14.2 - 31-32 A jar contains five red balls numbered 1 to...Ch. 14.2 - 31-32 A jar contains five red balls numbered 1 to...Ch. 14.2 - 33-40 Intersection of Events These exercises...Ch. 14.2 - 33-40 Intersection of Events These exercises...Ch. 14.2 - 33-40 Intersection of Events These exercises...Ch. 14.2 - 33-40 Intersection of Events These exercises...Ch. 14.2 - 33-40 Intersection of Events These exercises...Ch. 14.2 - 33-40 Intersection of Events These exercises...Ch. 14.2 - 39-40 Spinners A and B shown in the figure are...Ch. 14.2 - 39-40 Spinners A and B shown in the figure are...Ch. 14.2 - Four Siblings A couple intends to have four...Ch. 14.2 - Bridge Hands What is the probability that a...Ch. 14.2 - Roulette An American roulette wheel has 38 slots....Ch. 14.2 - Making Words A toddler has wooden blocks showing...Ch. 14.2 - Lottery In the 6/49 lottery game, a player selects...Ch. 14.2 - An unlikely event The president of a large company...Ch. 14.2 - Guessing on a test An exam has ten true-false...Ch. 14.2 - Quality Control To control the quality of their...Ch. 14.2 - Monkeys Typing Shakespeare An often-quoted example...Ch. 14.2 - Making Words A monkey is trained to arrange wooden...Ch. 14.2 - APPLICATIONS Making Words A toddler has eight...Ch. 14.2 - APPLICATIONS Horse Race Eight horses are entered...Ch. 14.2 - APPLICATIONS Genetics Many genetic traits are...Ch. 14.2 - APPLICATIONS Genetics Refer to Exercise 53. Make a...Ch. 14.2 - APPLICATIONS Roulette An American roulette wheel...Ch. 14.2 - APPLICATIONS Choosing a Committee A committee of...Ch. 14.2 - APPLICATIONS Snake eyes What is the probability of...Ch. 14.2 - APPLICATIONS Lottery In the 6/49 lottery game a...Ch. 14.2 - APPLICATIONS Marbles In a Jar A jar contains six...Ch. 14.2 - APPLICATIONS LotteryIn the 6/49 lottery game, a...Ch. 14.2 - APPLICATIONS Balls in a JarJar A contains three...Ch. 14.2 - APPLICATIONS Slot Machine A slot machine has three...Ch. 14.2 - APPLICATIONS Combination Lock A student has locked...Ch. 14.2 - APPLICATIONS Committee Membership A mathematics...Ch. 14.2 - APPLICATIONS Class Photo Twenty students are...Ch. 14.2 - APPLICATIONS Making Words A monkey is trained to...Ch. 14.2 - APPLICATIONS Making WordsA monkey is trained to...Ch. 14.2 - DISCUSS:Oldest SonA family with two children is...Ch. 14.3 - CONCEPTS A binomial experiment is one in which...Ch. 14.3 - CONCEPTS If a binomial experiment has probability...Ch. 14.3 - SKILLS 3-14Binomial Trials Five independent trials...Ch. 14.3 - SKILLS 3-14Binomial Trials Five independent trials...Ch. 14.3 - SKILLS 3-14Binomial Trials Five independent trials...Ch. 14.3 - SKILLS 3-14Binomial Trials Five independent trials...Ch. 14.3 - SKILLS 3-14Binomial Trials Five independent trials...Ch. 14.3 - SKILLS 3-14Binomial Trials Five independent trials...Ch. 14.3 - SKILLS 3-14Binomial Trials Five independent trials...Ch. 14.3 - SKILLS 3-14Binomial Trials Five independent trials...Ch. 14.3 - SKILLS 3-14 Binomial Trials Five independent...Ch. 14.3 - SKILLS 3-14 Binomial Trials Five independent...Ch. 14.3 - SKILLS 3-14 Binomial Trials Five independent...Ch. 14.3 - SKILLS 3-14 Binomial Trials Five independent...Ch. 14.3 - SKILLS 15-16 Probability Distribution An...Ch. 14.3 - SKILLS 15-16 Probability Distribution An...Ch. 14.3 - SKILLS 17-20 Probability Distribution A binomial...Ch. 14.3 - SKILLS 17-20 Probability Distribution A binomial...Ch. 14.3 - SKILLS 17-20 Probability Distribution A binomial...Ch. 14.3 - SKILLS 17-20 Probability Distribution A binomial...Ch. 14.3 - Rolling DiceSix dice are rolled. Find the...Ch. 14.3 - ArcheryAn archer hits his target 80 of the time....Ch. 14.3 - Television RatingsAccording to a ratings survey,...Ch. 14.3 - Spread of DiseaseHealth authorities estimate that...Ch. 14.3 - Blood TypeAbout 45 of the populations of the...Ch. 14.3 - HandednessA psychologist needs 12 left-handed...Ch. 14.3 - Germination RatesA certain brand of tomato seeds...Ch. 14.3 - Genders of ChildrenAssume that for any given line...Ch. 14.3 - Genders of ChildrenThe ratio of male to female...Ch. 14.3 - Education LevelIn a certain county 20 of the...Ch. 14.3 - Defective Light Bulbs The Dim Bulb Lighting...Ch. 14.3 - Quality Control An assembly line that manufactures...Ch. 14.3 - Sick Leave The probability that a given worker at...Ch. 14.3 - Political Surveys In a certain county, 60 of the...Ch. 14.3 - Pharmaceuticals A drug that is used to prevent...Ch. 14.3 - Reliability of a Machine A machine that is used in...Ch. 14.3 - Genetics Huntingtons disease is a hereditary...Ch. 14.3 - Selecting CardsThree cards are randomly selected...Ch. 14.3 - Smokers and Nonsmokers The participants at a...Ch. 14.3 - Telephone Marketing A mortgage company advertises...Ch. 14.3 - Effectiveness of a Drug A certain disease has a...Ch. 14.3 - Hitting a Target An archer normally hits the...Ch. 14.3 - DISCUSS: Most Likely Outcome for nTosses of a Coin...Ch. 14.4 - CONCEPTS 1. If a game gives payoffs of 10 and 100...Ch. 14.4 - CONCEPTS If you played the game in Exercise 1 many...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - SKILLS 3-12 Expected Value of a Game Find the...Ch. 14.4 - APPLICATIONS Roulette An American roulette wheel...Ch. 14.4 - APPLICATIONS Sweepstakes A sweepstakes offers a...Ch. 14.4 - APPLICATIONS A Game of Chance A box contains 100...Ch. 14.4 - APPLICATIONS Combination Lock A safe containing...Ch. 14.4 - APPLICATIONS Gambling on Stocks An investor buys...Ch. 14.4 - APPLICATIONS Slot Machine A slot machine has three...Ch. 14.4 - APPLICATIONS Lottery In a 6/49 lottery game, a...Ch. 14.4 - APPLICATIONS Lightning Insurance An insurance...Ch. 14.4 - APPLICATION Expected Number During the school year...Ch. 14.4 - APPLICATION Expected Number In a large liberal...Ch. 14.4 - APPLICATION Expected Number A student goes to swim...Ch. 14.4 - APPLICATION Expected Number Consider families with...Ch. 14.4 - APPLICATION 25-30 A fair Game? A game of chance...Ch. 14.4 - APPLICATIONS 25-30 A Fair Game? A game of chance...Ch. 14.4 - 25-30 A Fair Game? A game of chance is described....Ch. 14.4 - 25-30 A Fair Game? A game of chance is described....Ch. 14.4 - APPLICATIONS 25-30 A Fair Game?A game of chance...Ch. 14.4 - APPLICATIONS 25-30 A Fair Game?A game of chance...Ch. 14.4 - DISCUSSDISCOVERPROVEWRITE DISCUSS: The Expected...Ch. 14.CR - a What does the Fundamental Counting Principle...Ch. 14.CR - a What is a permutation of r elements of a set?...Ch. 14.CR - a What is a combination of r elements of a set?...Ch. 14.CR - a In solving a problem involving picking r objects...Ch. 14.CR - a What is meant by an experiment? Sample space? b...Ch. 14.CR - a What are mutually exclusive events? b If E and F...Ch. 14.CR - a What is meant by the conditional probability of...Ch. 14.CR - a An experiment has two outcomes, success and...Ch. 14.CR - a Suppose that a game gives payouts a1,a2,...,an...Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - 1-24 Counting These exercises involve counting....Ch. 14.CR - Probability These exercises involve probability....Ch. 14.CR - Marbles in a Jar If two marbles are drawn from the...Ch. 14.CR - Coin Toss A coin is tossed three times in a row,...Ch. 14.CR - Choosing Books A shelf has ten books: two...Ch. 14.CR - Rolling a Die and Selecting a Card A die is...Ch. 14.CR - Selecting Cards Find the probability that the...Ch. 14.CR - Rolling a Die, Selecting a Card, and tossing a...Ch. 14.CR - Rolling Dice Two dice are rolled. Find the...Ch. 14.CR - Selecting a Card Four cards are dealt from a...Ch. 14.CR - Lottery In the numbers game lottery a player picks...Ch. 14.CR - Game Show In a TV game show, a contestant is given...Ch. 14.CR - 25-42 ProbabilityThese exercises involve...Ch. 14.CR - 25-42 ProbabilityThese exercises involve...Ch. 14.CR - 25-42 ProbabilityThese exercises involve...Ch. 14.CR - 25-42 ProbabilityThese exercises involve...Ch. 14.CR - 25-42 ProbabilityThese exercises involve...Ch. 14.CR - 25-42 ProbabilityThese exercises involve...Ch. 14.CR - 25-42 ProbabilityThese exercises involve...Ch. 14.CR - 43-46 Binomial ProbabilityThese exercises involve...Ch. 14.CR - 43-46 Binomial ProbabilityThese exercises involve...Ch. 14.CR - 43-46 Binomial ProbabilityThese exercises involve...Ch. 14.CR - 43-46 Binomial ProbabilityThese exercises involve...Ch. 14.CR - 47-50Expected ValueThese exercises involve...Ch. 14.CR - 47-50Expected ValueThese exercises involve...Ch. 14.CR - 47-50Expected ValueThese exercises involve...Ch. 14.CR - 47-50Expected ValueThese exercises involve...Ch. 14.CT - Alice and Bill have four grandchildren, and they...Ch. 14.CT - A hospital cafeteria offers a fixed-price lunch...Ch. 14.CT - An Internet service provider requires its customer...Ch. 14.CT - Over the past year, John has purchased 30 books. a...Ch. 14.CT - A commuter must travel from Ajax to Barrie and...Ch. 14.CT - A pizza parlor offers four sizes of pizza and 14...Ch. 14.CT - An anagram of a word is a rearrangement of the...Ch. 14.CT - A board of directors consisting of eight members...Ch. 14.CT - One card is drawn from a deck. Find the...Ch. 14.CT - A jar contains five red balls, numbered 1 to 5,...Ch. 14.CT - Three people are chosen at random from a group of...Ch. 14.CT - 12. Two dice are rolled. What is the probability...Ch. 14.CT - 13. In a group of four students, what is the...Ch. 14.CT - An unbalanced coin weighted so that the...Ch. 14.CT - You are to draw one card from a deck. If it is an...Ch. 14.FOM - 1. Winning Strategy In a game show like the one...Ch. 14.FOM - Family PlanningA couple intends to have two...Ch. 14.FOM - Dividing a JackpotA game between two players...Ch. 14.FOM - Estimating In this problem we use the Monte Carlo...Ch. 14.FOM - Areas of Curved Regions The Monte Carlo method can...Ch. 14.FOM - Random NumbersChoose two numbers at random from...

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