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Counting - Cengage

Countingand ProbabilityProbability is the mathematicalstudy of chance and randomprocesses. The laws of probabilityare essential for understandinggenetics, opinion polls, pricingstock options, setting odds inhorseracing and games ofchance, and many other questions in mathematics involve Counting . For example, in how many wayscan a committee of two men and three women be chosen from a group of 35 menand 40 women? How many different license plates can be made using three lettersfollowed by three digits? How many different poker hands are possible?Closely related to the problem of Counting is that of probability. We considerquestions such as these: If a committee of five people is chosen randomly from agroup of 35 men and 40 women, what are the chances that no women will be cho-sen for the committee? What is the likelihood of getting a straight flush in a pokergame?

Counting and Probability Probability is the mathematical study of chance and random processes. ... An all-star baseball team has a roster of seven pitchers and

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1 Countingand ProbabilityProbability is the mathematicalstudy of chance and randomprocesses. The laws of probabilityare essential for understandinggenetics, opinion polls, pricingstock options, setting odds inhorseracing and games ofchance, and many other questions in mathematics involve Counting . For example, in how many wayscan a committee of two men and three women be chosen from a group of 35 menand 40 women? How many different license plates can be made using three lettersfollowed by three digits? How many different poker hands are possible?Closely related to the problem of Counting is that of probability. We considerquestions such as these: If a committee of five people is chosen randomly from agroup of 35 men and 40 women, what are the chances that no women will be cho-sen for the committee? What is the likelihood of getting a straight flush in a pokergame?

2 In studying probability, we give precise mathematical meaning to phrasessuch as what are the chances .. ? and what is the likelihood .. ? Suppose that three towns, Ashbury, Brampton, and Carmichael, are located in sucha way that two roads connect Ashbury to Brampton and three roads connectBrampton to Carmichael. How many different routes can one take to travel fromAshbury to Carmichael via Brampton? The key idea in answering this question is toconsider the problem in stages. At the first stage from Ashbury to Brampton there are two choices. For each of these choices, there are three choices to make atthe second stage from Brampton to Carmichael. Thus, the number of differentroutes is 2 3 6. These routes are conveniently enumerated by a tree diagramas in Figure method used to solve this problem leads to the following is an immediate consequence of this principle for any number of events: IfE1, E2.

3 , Ekare events that occur in order and if E1can occur in n1ways, E2in n2ways, and so on, then the events can occur in order in n1 n2 .. 1 Using the Fundamental Counting PrincipleAn ice-cream store offers three types of cones and 31 flavors. How many differentsingle-scoop ice-cream cones is it possible to buy at this store?FUNDAMENTAL Counting PRINCIPLES uppose that two events occur in order. If the first can occur in mways andthe second in nways (after the first has occurred), then the two events canoccur in order in m Counting PRINCIPLESWhen it is not in our power to determine what is true,we ought to follow what is most DESCARTESA shburyBramptonCarmichaelpqzyxFIGURE 1 Tree diagramABBCCCCCCR outeqpxxyyzzpxpypzqxqyqzSOLUTIONT here are two choices: type of cone and flavor of ice cream. At the first stage wechoose a type of cone, and at the second stage we choose a flavor.

4 We can think ofthe different stages as boxes: stage 1 stage 2type of cone flavorThe first box can be filled in three ways and the second in 31 ways: stage 1 stage 2 Thus, by the Fundamental Counting Principle, there are 3 31 93 ways ofchoosing a single-scoop ice-cream cone at this store. EXAMPLE 2 Using the Fundamental Counting PrincipleIn a certain state, automobile license plates display three letters followed by threedigits. How many such plates are possible if repetition of the letters(a) is allowed?(b) is not allowed?SOLUTION(a) There are six choices, one for each letter or digit on the license plate. As in thepreceding example, we sketch a box for each stage:lettersdigitsAt the first stage, we choose a letter (from 26 possible choices); at the secondstage, another letter (again from 26 choices); at the third stage, another letter(26 choices); at the fourth stage, a digit (from 10 possible choices); at the fifthstage, a digit (again from 10 choices); and at the sixth stage, another digit (10choices).

5 By the Fundamental Counting Principle, the number of possiblelicense plates is26 26 26 10 10 10 17,576,000(b) If repetition of letters is not allowed, then we can arrange the choices asfollows:lettersdigits26252410101026262 6101010331868 CHAPTER 11 Counting and ProbabilityPersi Diaconis(b. 1945) is cur-rently professor of statistics andmathematics at Stanford Universityin California. He was born in NewYork City into a musical familyand studied violin until the age of14. At that time he left home tobecome a magician. He was amagician (apprentice and master)for ten years. Magic is still hispassion, and if there were a profes-sorship for magic, he would cer-tainly qualify for such a post! Hisinterest in card tricks led him to astudy of probability and is now one of the leading statis-ticians in the world. With his back-ground he approaches mathematicswith an undeniable flair.

6 He says Statistics is the physics of num-bers. Numbers seem to arise in theworld in an orderly fashion. Whenwe examine the world, the sameregularities seem to appear againand again. Among his many origi-nal contributions to mathematics isa probabilistic study of the perfectcard the first stage, we have 26 letters to choose from, but once the first letter ischosen, there are only 25 letters to choose from at the second stage. Once thefirst two letters are chosen, 24 letters are left to choose from for the thirdstage. The digits are chosen as before. Thus, the number of possible licenseplates in this case is26 25 24 10 10 10 15,600,000 EXAMPLE 3 Using Factorial NotationIn how many different ways can a race with six runners be completed? Assumethere is no are six possible choices for first place, five choices for second place (sinceonly five runners are left after first place has been decided), four choices for thirdplace, and so on.

7 So, by the Fundamental Counting Principle, the number ofdifferent ways this race can be completed is 6 5 4 3 2 1 6! 720 SECTION Principles869 Factorial notation is explained on page vendor sells ice cream from a cart on the boardwalk. Heoffers vanilla, chocolate, strawberry, and pistachio icecream, served on either a waffle, sugar, or plain cone. Howmany different single-scoop ice-cream cones can you buyfrom this vendor? many three-letter words (strings of letters) can beformed using the 26 letters of the alphabet if repetition ofletters(a)is allowed?(b)is not allowed? many three-letter words (strings of letters) can beformed using the letters WXYZif repetition of letters(a)is allowed?(b)is not allowed? horses are entered in a race.(a)How many different orders are possible for completingthe race?(b)In how many different ways can first, second, and thirdplaces be decided?

8 (Assume there is no tie.) multiple-choice test has five questions with four choicesfor each question. In how many different ways can the testbe completed? numbers consist of seven digits; the first digitcannot be 0 or 1. How many telephone numbers arepossible? how many different ways can a race with five runners becompleted? (Assume there is no tie.) how many ways can five people be seated in a row offive seats? restaurant offers six different main courses, eight typesof drinks, and three kinds of desserts. How many differentmeals consisting of a main course, a drink, and a dessertdoes the restaurant offer? how many ways can five different mathematics books beplaced next to each other on a shelf ? A, B, C, and D are located in such a way that thereare four roads from A to B, five roads from B to C, and sixroads from C to D. How many routes are there from townA to town D via towns B and C?

9 A family of four children, how many different boy-girlbirth-order combinations are possible? (The birth ordersBBBGand BBGBare different.) coin is flipped five times, and the resulting sequence ofheads and tails is recorded. How many such sequences arepossible? red die and a white die are rolled, and the numbersshowing are recorded. How many different outcomes arepossible? (The singular form of the word diceis die.) red die, a blue die, and a white die are rolled, and thenumbers showing are recorded. How many different out-comes are possible? cards are chosen in order from a deck. In how manyways can this be done if(a)the first card must be a spade and the second must be aheart?(b)both cards must be spades? girl has 5 skirts, 8 blouses, and 12 pairs of shoes. Howmany different skirt-blouse-shoe outfits can she wear?(Assume that each item matches all the others, so she iswilling to wear any combination.)

10 Company s employee ID number system consists of oneletter followed by three digits. How many different IDnumbers are possible with this system? company has 2844 employees. Each employee is to begiven an ID number that consists of one letter followed bytwo digits. Is it possible to give each employee a differentID number using this scheme? all-star baseball team has a roster of seven pitchers andthree catchers. How many pitcher-catcher pairs can themanager select from this roster? automobile license plates in California display anonzero digit, followed by three letters, followed by threedigits. How many different standard plates are possible inthis system? combination lock has 60 different positions. To open thelock, the dial is turned to a certain number in the clockwisedirection, then to a number in the counterclockwise direc-tion, and finally to a third number in the clockwise direc-tion.


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