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All Textbook Solutions for Finite Mathematics for Business, Economics, Life Sciences and Social Sciences

17E 18E 19E 20E In Problems 17-24, use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. nA=110,nB=220,nAB=60,nU=300 In Problems 17-24, use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. nA=70,nB=170,nAB=40,nU=300 In Problems 17-24, use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. nA=20,nB=40,nAB=50,nU=80 In Problems 17-24, use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. nA=30,nB=10,nAB=35,nU=60 In Problems 25-32, use the given information to complete the following table, AATotalsB???B???Totals??? nA=70,nB=90,nAB=30,nU=200 In Problems 25-32, use the given information to complete the following table, AATotalsB???B???Totals??? nA=55,nB=65,nAB=35,nU=100 In Problems 25-32, use the given information to complete the following table, AATotalsB???B???Totals??? nA=45,nB=55,nAB=80,nU=100 In Problems 25-32, use the given information to complete the following table, AATotalsB???B???Totals??? nA=80,nB=70,nAB=110,nU=200 In Problems 25-32, use the given information to complete the following table, AATotalsB???B???Totals??? nA=15,nB=24,nAB=32,nU=90 In Problems 25-32, use the given information to complete the following table, AATotalsB???B???Totals??? nA=81,nB=90,nAB=63,nU=180 In Problems 25-32, use the given information to complete the following table, AATotalsB???B???Totals??? nA=110,nB=145,nAB=255,nU=300 In Problems 25-32, use the given information to complete the following table, AATotalsB???B???Totals??? nA=175,nB=125,nAB=300,nU=300 In Problems 33 and 34, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. A If AorB is the empty set, then AorB are disjoint. B If AorB are disjoint, then AorB is the empty set.In Problems 33 and 34, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. A If AorB are disjoint, then nAB=nA+nB. B If nAB=nA+nB, then AorB are disjoint.A particular new car model is available with 5 choices of color, 3 choices of transmission, 4 types of interior, and 2 types of engine. How many different variations of this model are possible?A delicatessen serves meat and sandwich with the following options: 3 kinds of bread, 5 kinds of meat, and lettuce or sprouts. How many different sandwiches are possible, assuming that on item is used out of each category?Using the English alphabet, how many 5 -character case sensitive passwords are possible?Using the English alphabet, how many 5 -character case sensitive passwords are possible if each character is a letter or a digit?A combination lock has 5 wheels, each labeled with the 10 digits from 0to9. How many opening combinations are possible if no digit is repeated? If digits ca be repeated? If successive digits must be different?A combination lock has 3 wheels, each labeled with the 10 digits from 0to9. How many 3 -digit combinations are possible if no digit is repeated? If digits can be repeated? If successive digits must be different?How many different license plates are possible if each contains 3 letters (out of the alpabet’s 26 letters) followed by 3 digits (from 0to9 )? How many these license plates contain no repeated letters and no repeated digits?How many 5 digit ZIP code numbers are possible? How many of these numbers contain no repeated digits?43E 44E Problems 45-48 refer to the following Venn diagram. Which of the numbers x,y,z,orw must equal to 0 if BA ?Problems 45-48 refer to the following Venn diagram. Which of the numbers x,y,z,orw must equal to 0 if AandB are disjoint?Problems 45-48 refer to the following Venn diagram. Which of the numbers x,y,z,orw must equal to 0 if AB=AB ?Problems 45-48 refer to the following Venn diagram. Which of the numbers x,y,z,orw must equal to 0 if AB=U ?A group of 75 people includes 32 who play tennis, 37 who play golf, and 8 who play both tennis and golf. How many people in the group play neither sport?A class of 30 music student includes 13 who play the piano, 16 who play the guitar, and 5 who play both the piano and the guitar. How many students in the class play neither instrument?A group of 100 people touring Europe includes 42 people who speak French, 55 who speak German, and 17 who speak neither language. How many in the group speak both French and German?A high school football team with 40 players includes 16 players who were played offense last year, 17 who played defense, and 12 who were not on last year’s team. How many players from last year played both offense and defense?53E Management. A corporation plans to fill 2 positions for vice-president, V1 and V2, from administrative officers in 2 of its manufacturing plants. Plant A has 6 off and plant B has 8. How many ways can these 2 positions filled if the V1 position is to be filled from plant A and the position from plant B ? How many ways can these two positions be filed if the selection is made without regard to plant?Transportation, A sales representative who lives in city A wishes to start from home and visit 3 different cities: B,C,andD. She must choose whether to drive her own car or to fly. If all cities are interconnected by both roads and airlines, how many travel plans can be constructed to visit each city exactly once and return home?Transportation. A sales representative who lives in city A wishes to start from home and visit 4 different cities: B,C,D,andE. If road interconnects all 4 cities, how many different route plans can be constructed so that a single truck, starting from A, will visit each city exactly once, then return home?Market research. A survey of 1,200 people indicates that 850 own HDTVs, 740 own DVD players, and 580 own HDTVs and DVD players. A How many people in the survey own either HDTV or a DVD player? B How many own neither an HDTV nor a DVD player? C How many own an HDTV and do not own a DVD player?Market research. A survey of 800 small businesses indicates that 250 own a video conferencing system, 420 own projection equipment, and 180 own a video conferencing and projection equipment. A How many businesses in the survey own either video conferencing or a projection equipment? B How many own neither an video conferencing nor a projection equipment? C How many own projection equipment and do not own a video conferencing system?Communications. A cable television company has 8,000 subscribers in a suburban community. The company offers two premium channels: HBO and Showtime. If 2,450 subscribers receive HBO, 1,940 receive Showtime, and 5,180 do not receive any premium channel. How many subscribers receive both HBO and Showtime?Communications. A cable company offers its 10,000 two special services: high speed internet and digital phone. If 3,770 customers use high speed internet, 3,250 use digital phone, and 4,530 do not use either of these services, how many customers use both high-speed internet and digital phone?Minimum wage. The table gives the number of male and female workers earning at or below the minimum wage for several age categories? A How many males are of age 20-24 and earn below minimum wage? B How many females are of age 20 or older and earn below minimum wage? C How many workers are of age 16-19 or males and earning below minimum wage? D How many workers earn below minimum wage?Minimum wage. Refer to table in Problem 61. A How many females are of age 16-19 and earn below minimum wage? B How many males are of age 16-24 and earn below minimum wage? C How many workers are of age 20-24 or females and earning below minimum wage? D How many workers earn minimum wage?Medicine. A medical researcher classifies subjects according to male or female; smoker or nonsmoker; and underweight, average weight, or overweight. How many combined classifications are possible? A Solve using tree diagram. B Solve using the multiplication principle.Family Planning. A couple is planning to have 3 children. How my boy-girl combinations are possible? Distinguish between combined outcomes such as B,B,G,B,G,B,andG,B,B. A Solve using tree diagram. B Solve using the multiplication principle.Politics. A politician running for a third term is planning to contact all contributors to her first two campaigns. If 1,475 individuals contributed to the first campaign, 2,350 contributed to the second campaign, and 920 contributed to the first and second campaigns, how many have contributed to the first or second campaign?Politics. If 12,457 people voted for a politician in his first election, 15,322 voted for him in his second election, and 9,345 voted for him in the first and second elections, how many people have voted for this politician in the first or second election?A List alphabetically by the first letter, all 3letter license plate codes consisting of 3 different letters chosen from M,A,T,H. Discuss how this list relates to nPr. B Recognize the list from A so that all codes without M comes first, then all codes without A, then all codes without T, and finally all codes without H. Discuss how this illustrates the formula nPr=r!nCr.Find A6!B10!9!C10!7!D5!0!3!E20!3!17!Given the set A,B,C,D, how many permutations are possible for this set of 4 objects taken 2 at a time? Answer the question (A) Using the diagram (B) Using the multiplication principle (C) Using the two formula for Pnr Find the number of permutations of 30 objects taken 4 at a time. Compute the answer using a calculator.From a committee of 12 people, (A) In how many ways can be choose a chairperson, a vice-chairperson, a sectary, and a treasurer, assuming that one person cannot hold more than one position? (B) In how many ways can we choose a subcommittee of 4 people?Find the number of combinations of 30 objects taken 4 at a time. Compute the answer using a calculator,How many 5-card hands have 3 hearts and 2 spades?Repeat Example 7 under the same conditions, except that the serial numbers will now have 3 letters followed by 2 digits (no repeats).Given the information in Example 8, answer the following questions: A How many 4-officer committees with 1 senior officer and 3 junior officers can be formed? B How many 4-officer committees with 4 junior officers can be formed? C How many 4-officer committees with at least 2 junior officers can be formed?From a standard 52-card deck, how many 5-card hands have all cards from the same suit?In Problems 1-6, evaluate the given expression without using a calculator. (If necessary, review Section A.4). 121110321 In Problems 1-6, evaluate the given expression without using a calculator. (If necessary, review Section A.4). 12108642 In Problems 1-6, evaluate the given expression without using a calculator. (If necessary, review Section A.4). 10987654321 In Problems 1-6, evaluate the given expression without using a calculator. (If necessary, review Section A.4). 87654321 In Problems 1-6, evaluate the given expression without using a calculator. (If necessary, review Section A.4). 1009998...321989796...321 In Problems 1-6, evaluate the given expression without using a calculator. (If necessary, review Section A.4). 11109...32187654321 In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 8!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 9!9E In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 7+3!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 2317!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 5247!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 11!8!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 20!18!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 8!4!84!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 10!5!105!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 500!498!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 601!599!In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 13C8 In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 15C10 In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 18P6 In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 10P7 In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 12P7127 In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 365P2536525 In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 39C552C5 In Problems 7-26, evaluate the expression. If the answer is not an integer, round to four decimal places. 26C452C4 In Problems 27-30, simplify each expression assuming that n is an integer and n2. n!n2!In Problems 27-30, simplify each expression assuming that n is an integer and n2. n+1!3!n2!In Problems 27-30, simplify each expression assuming that n is an integer and n2. n+1!2!n1!In Problems 27-30, simplify each expression assuming that n is an integer and n2. n+3!n+1!In Problems 31-36, would you consider the selection to be a permutation, a combination, or neither? Explain your reasoning. The university president named 3 new officers: a vice-president of finance, a vice-president of academic affairs, and a vice-president of student affairs.In Problems 31-36, would you consider the selection to be a permutation, a combination, or neither? Explain your reasoning. The university president selected 2 of her a vice-presidents to attend the dedication ceremony of a new branch campus.In Problems 31-36, would you consider the selection to be a permutation, a combination, or neither? Explain your reasoning. A student checked out 4 novels from the library.In Problems 31-36, would you consider the selection to be a permutation, a combination, or neither? Explain your reasoning. A student bought 4 books: 1 for his father, 1 for his mother, 1 for his younger sister, and 1 for his older brother.In Problems 31-36, would you consider the selection to be a permutation, a combination, or neither? Explain your reasoning. A father ordered an ice-cream cone (chocolate, vanilla, or strawberry) for each of his 4 children.In Problems 31-36, would you consider the selection to be a permutation, a combination, or neither? Explain your reasoning. A book club meets monthly at the home of one of its 10 members. In December, the club selects a host for each meeting of the next year.In a horse race, how many different finishes among the first 3 places are possible if 10 horses are running? (Exclude ties.)In a long-distance foot race, how many different finishes among the first 5 places are possible if 50 people are running? (Exclude ties.)How many ways can a 3 -person subcommittee be selected from a committee of 7 people? How many ways can a president, vice-president, and secretary be chosen from committee of 7 people?Nine cards are numbered with digits from 1 to 9. A 3 -card hand is dealt, 1 card at a time. How many hands are possible in which A Order is taken into consideration? B Order is not taken into consideration?From a standard 52 -card deck, how many 6 -card hands consist entirely of red cards?From a standard 52 -card deck, how many 6 -card hands consist entirely of clubs?From a standard 52 -card deck, how many 5 -card hands consist entirely of face cards?From a standard 52 -card deck, how many 5 -card hands consist entirely of queens?From a standard 52 -card deck, how many 7 -card hands consist entirely of four kings?From a standard 52 -card deck, how many 7 -card hands consist 3 hearts and 4 diamonds?From a standard 52 -card deck, how many 4 -card hands contain a card from each suit?From a standard 52 -card deck, how many 4 -card hands contain a card from same suit?From a standard 52 -card deck, how many 5 -card hands contain 3 cards of one rank and 2 cards of a different rank?From a standard 52 -card deck, how many 5 -card hands contain 5 different ranks of cards?A catering service offers 8 appetizers, 10 main courses, and 7 desserts. A banquet committee selects 3 appetizers, 4 main courses, and 2 desserts. How many ways can this be done?Three departments have 12,15,and18 members respectively. If each department selects a delegate and an alternate to represent the department at a conference, how many ways can this be done?53E 54E In Problems 55-60, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample. If n is a positive integer, then n!n+1!In Problems 55-60, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample. If n is a positive integer greater than 3, then n!2n In Problems 55-60, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample. If n and r are positive integers and 1rn, then nPrnPr+1.In Problems 55-60, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample. If n and r are positive integers and 1rn, then nCrnCr+1.In Problems 55-60, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample. If n and r are positive integers and 1rn, then nCrnCnr.In Problems 55-60, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample. If n and r are positive integers and 1rn, then nPrnPnr.Eight distinct points are selected on the circumference of a circle. A How many line segments can be drawn by joining the points in all possible ways? B How many triangles can be drawn by using these 8 points as vertices? C How many quadrilaterals can be drawn by using these 8 points as vertices?Five distinct points are selected on the circumference of a circle. A How many line segments can be drawn by joining the points in all possible ways? B How many triangles can be drawn by using these 5 points as vertices? C How many quadrilaterals can be drawn by using these 5 points as vertices?In how many ways can 4 people sit in arrow of 6 chairs?In how many ways can 3 people sit in arrow of 7 chairs?A basketball team has 5 distinct positions. Out of 8 players, how many starting teams are possible if A The distinct positions are taken into consideration? B The distinct positions are not taken into consideration? C The distinct positions are not taken into consideration, but either Mike or Ken (but not both) must start?How many 4-person committees are possible from a group of 9 people if A There are no restrictions? B Both Jim and Mary must be on the committee? C Either Jim and Mary (but not both) must be on the committee?Let U be the set of all 2 -card hands, let K be the set of all 2 -card hands that contain exactly 1 king, and let H be the set of all 2 -card hands that contain exactly 1 heart. Find nKH,nKH,nKH,andnKH.Let U be the set of all 2 -card hands, let K be the set of all 2 -card hands that contain exactly 1 king, and let Q be the set of all 2 -card hands that contain exactly 1 queen. Find nKQ,nKQ,nKQ,andnKQ.69E Note from the table in the graphing calculator display below that the largest value of nCr when n=21 is 21C10=21C11=352,716. Use a similar table to find the largest value of nCr when n=17.Quality Control. An office supply store receives a shipment of 24 high speed printers, including 5 that are defective. Three of these printers are selected for a store display. A How many selections can be made? B How many of these selections will contain no defective printers?Quality Control. An electronics store receives a shipment of 30 graphing calculators, including 6 that are defective. Four of these calculators are selected for a local high school. A How many selections can be made? B How many of these selections will contain no defective printers?Business closings. A jewelry store chain with 8 stores in Georgia, 12 in Florida, and 10 in Alabama is planning to close 10 of these stores. A How many ways can this be done? B The company decides to close 2 stores in Georgia, 5 in Florida, and 3 in Alabama. In how many ways can this be done?Employee layoffs. A real estate company with 14 employees in its central office, 8 in its north office, and 6 in its south office is planning to lay off 12 employees. A How many ways can this be done? B The company decides to lay off 5 employees from the central office, and 4 from the north office, and 3 from the south office. In how many ways can this be done?Personal selection. Suppose that 6 female and 5 male applicants have been successfully screened for 5 positions. In how many ways can the following composition be selected? A 3 females and 2 males B 4 females and 1 males C 5 females D 5 people regardless of sex E At least 4 females A 4 -person grievance committee is selected out of 2 departments AandB, with 15and20 people, respectively. In how many ways can the following committees be selected? A 3 from A and 1 from B B 2 from A and 2 from B C All from A D 4 people regardless of department E At least 3 from A Medicine. There are 8 standard classifications of blood type. An examination for prospective laboratory technicians consists of having each candidate determine the type for 3 blood samples. How many different examinations can be given if no 2 of the samples provided for the candidate have the same type? If 2 or more samples have the same type?Medical research. Because of limited funds, 5 research centers are chosen out of 8 suitable ones for a study on heart disease. How many choices are possible?Politics. A nominating convention will select a president and vice-president from among 4 candidates. Campaign buttons, listing a president and a vice-president, will be designed for each possible outcome before the convention. How many different kinds of buttons should be designed?Politics. In how many different ways can 6 candidates for an office be listed on a ballot?In Problems 1-6, express each proposition in an English sentence and determine whether it is true or false, where p and q are the propositions p:2332q:3443 q In Problems 1-6, express each proposition in an English sentence and determine whether it is true or false, where p and q are the propositions p:2332q:3443 pq In Problems 1-6, express each proposition in an English sentence and determine whether it is true or false, where p and q are the propositions p:2332q:3443 pq In Problems 1-6, express each proposition in an English sentence and determine whether it is true or false, where p and q are the propositions p:2332q:3443 pq In Problems 1-6, express each proposition in an English sentence and determine whether it is true or false, where p and q are the propositions p:2332q:3443 The converse of pq In Problems 1-6, express each proposition in an English sentence and determine whether it is true or false, where p and q are the propositions p:2332q:3443 The contrapositive of pq In Problems 7-10, indicate true T or false F, 5,6,7=6,7,5 In Problems 7-10, indicate true T or false F, 555,555 In Problems 7-10, indicate true T or false F, 9,273,9,27,81 In Problems 7-10, indicate true T or false F, 1,21,1,2 In Problems 11-14, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true or false. If 9 is prime, then 10 is odd.In Problems 11-14, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true or false. 7 is even or 8 is odd.In Problems 11-14, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true or false. 53 is prime and 57 is prime.In Problems 11-14, describe each proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true or false. 51 is not prime.In Problems 15-16, state the converse and the contrapositive of the given proposition. If the square matrix A has a row of zeroes, then the square matrix A has no inverse.In Problems 15-16, state the converse and the contrapositive of the given proposition. If the square matrix A is an identity matrix, then the square matrix A has an inverse.In Problems 17-19, write the resulting set using the listing method. 1,2,3,42,3,4,5 In Problems 17-19, write the resulting set using the listing method. 1,2,3,42,3,4,5 In Problems 17-19, write the resulting set using the listing method. 1,2,3,45,6 20RE 21RE 22RE Evaluate the expressions in Problems 23-28. 106!Evaluate the expressions in Problems 23-28. 15!10!Evaluate the expressions in Problems 23-28. 15!10!5!Evaluate the expressions in Problems 23-28. C85 Evaluate the expressions in Problems 23-28. P85 Evaluate the expressions in Problems 23-28. C134C131 How many seating arrangements are possible with 6 people and 6 chairs in a row? Solve using the multiplication principle.Solve Problem 29 using permutations or combinations, whichever is applicable.In Problems 31-36, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqqp In Problems 31-36, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pqp In Problems 31-36, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. ppqq In Problems 31-36, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. qpq In Problems 31-36, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. ppq In Problems 31-36, construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. pq In Problems 37-40, determine whether the given set is finite or infinite. Consider the set Z of integers to be the universal set, and let M=nZn106K=nZn103E=nZniseven EK In Problems 37-40, determine whether the given set is finite or infinite. Consider the set Z of integers to be the universal set, and let M=nZn106K=nZn103E=nZniseven MK 39RE In Problems 37-40, determine whether the given set is finite or infinite. Consider the set Z of integers to be the universal set, and let M=nZn106K=nZn103E=nZniseven EM In Problems 41-42, determine whether or not the given sets are disjoint. For the definitions of M,K, and E, refer to the instructions for Problems 37-40. MandK In Problems 41-42, determine whether or not the given sets are disjoint. For the definitions of M,K, and E, refer to the instructions for Problems 37-40. MandE 43RE A man has 5 children. Each of those children has 3 children, who in turn each have 2 children. Discuss the number of descendants that the man has.How many 3-letter code words are possible using the first 8 letters of the alphabet if no letter can be repeated? If letters can be repeated? If adjacent letters cannot be alike?Solve the following problems using Pnr or Cnr : (A) How many 3-digit opening combinations are possible on a combination lock with 6 digits if the digits cannot be repeated? (B) Five tennis players have made the finals. If each of the 5 players is to play every other player exactly once, how many games must be scheduled?Use graphical techniques on a graphing calculator to find the largest value of Cnr when n=25.48RE In Problems 49-51, write the resulting set using the listing method. xx3x=0 In Problems 49-51, write the resulting set using the listing method. xxisapositiveintegerandx!100 In Problems 49-51, write the resulting set using the listing method. xxisapositiveintegerthatisaperfectsquareandx50 A software development department consists of 6 women and 4 men. (A) How many ways can the department select a chief programmer, a backup programmer, and a programming librarian? (B) How many of the selections in part (A) consists entirely of women? (C) How many ways can the department select a team of 3 programmers to work on a particular project?A group of 150 people includes 52 who plays chess, 93 who play checkers, and 28 who play both chess and checkers. How many people in the group play neither game?Problems 54 and 55 refer to the following Venn diagram. Which of the numbers x,y,z, or w must equal to 0 if AB ?Problems 54 and 55 refer to the following Venn diagram. Which of the numbers x,y,z, or w must equal to 0 if AB=U ?In Problems 56-58, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample. If n and r are positive integers and 1rn, then CnrPnr.In Problems 56-58, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample. If n and r are positive integers and 1rn, then Pnrn!.In Problems 56-58, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample. If n and r are positive integers and 1rn, then Cnrn!.In Problems 59-64, construct a truth table to verify the implication or equivalence. pqp In Problems 59-64, construct a truth table to verify the implication or equivalence. qpq In Problems 59-64, construct a truth table to verify the implication or equivalence. pqqp In Problems 59-64, construct a truth table to verify the implication or equivalence. pqpq In Problems 59-64, construct a truth table to verify the implication or equivalence. ppqq In Problems 59-64, construct a truth table to verify the implication or equivalence. pqpq How many different 5-child families are possible where the gender of the children in the order of their births is taken into consideration [that is, birth sequences such as B,G,G,B,B and G,B,G,B,B produce different families]? How many families are possible if the order pattern is not taken into account?Can a selection of r objects from a set of n distinct objects, where n is a positive integer, be a combination and a permutation simultaneously? Explain.Transportation. A distribution center A wishes to send its products to five different retail stores: B,C,D,E, and F. How many different route plans can be constructed so that a single truck, starting from A, will deliver to each store exactly once and then return to the center?Market research. A survey of 1,000 people indicates that 340 have invested in stocks, 480 have invested in bonds, and 210 have invested in stocks and bonds. (A) How many people in the survey have invested in stocks or bonds? (B) How many have invested in neither stocks nor bonds? (C) How many have invested in bonds and not stocks?Medical research. In a study of twins, a sample of 6 pairs of identical twins will be selected for medical tests from a group of 40 pairs of identical twins. In how many ways can this be done?Elections. In an unusual recall election, there are 67 candidates to replace the governor of a state. To negate the advantage that might accrue to candidates whose names appear near the top of the ballot, it is proposed that equal number of ballots be printed for each possible order in which the candidates’ names can be listed. (A) In how many ways can the candidates’ names be listed? (B) Explain why the proposal is not feasible, and discuss possible alternatives.A shipment box contains 12 graphing calculators, out of which 2 are defective. A calculator is drawn at random from the box and then, without replacement, a second calculator is drawn. Discuss whether the equally likely assumption would be appropriate for the sample space S=GG,GD,DG,DD where G is a good calculator and D is a defective one.Repeat Example 1 for (A) The outcome is a number divisible by 12. (B) The outcome is an even number greater than 15.An experiment consists of recording the boy-girl composition of a two-child family. What would be an appropriate sample space (A) If we are interested in the genders of the children in the order of their births? Draw a tree diagram. (B) If we are interested only in the number of girls in a family? (C) If we are interested only in whether the genders are alike (A) or different (D)? (D) For all three interests expressed in parts (A) to (C)?Refer to the sample space shown in Figure 2. What is the event that corresponds to each of the following outcomes? (A) A sum of 5 turns up. (B) A sum that is a prime number greater than 7 turns up.Suppose in Example 4 that after flipping the nickel and dime 1,000 times, we find that HH turns up 273 times, HT turns up 206 times, TH turns up 312 times, and TT turns up 209 times. On the basis of this evidence, we assign probabilities to the simple events in S as follows: This is an acceptable and reasonable probability assignment for the simple events in S. What are the probabilities of the following events? AE1=gettingatleast1tail. BE2=getting2tails. CE3=gettingatleast1headoratleast1tail Under the conditions in Example 5, find the probabilities of the following events (each event refers to the sum of the dots facing up on both dice): (A) E5=asumof5turnsup. (B) E6= a sum that is a prime number greater than 7 turns up.Use the graphing calculator output in Figure 4B to determine the empirical probabilities of the following events, and compare with the theoretical probabilities: AE3=asumlessthan4turnsup BE4=asumof12turnsup In drawing 7 cards from a 52 -card deck without replacement, what is the probability of getting 7 hearts?What is the probability that the committee in Example 8 will have 4 men and 2 women?In Problems 1-6, without using a calculator, determine which event, E or F, is more likely to occur. (If necessary, review Section A.l.) PE=56;PF=45 In Problems 1-6, without using a calculator, determine which event, E or F, is more likely to occur. (If necessary, review Section A.l.) PE=27;PF=13 In Problems 1-6, without using a calculator, determine which event, E or F, is more likely to occur. (If necessary, review Section A.l.) PE=38;PF=.4 In Problems 1-6, without using a calculator, determine which event, E or F, is more likely to occur. (If necessary, review Section A.l.) PE=.9;PF=78 In Problems 1-6, without using a calculator, determine which event, E or F, is more likely to occur. (If necessary, review Section A.l.) PE=.15;PF=16 In Problems 1-6, without using a calculator, determine which event, E or F, is more likely to occur. (If necessary, review Section A.l.) PE=57;PF=611 A circular spinner is divided into 15 sectors of equal area: 6 red sectors, 5 blue, 3 yellow, and 1 green. In Problems, 7-14, consider the experiment of spinning the spinner once. Find the probability that the spinner lands on: Blue A circular spinner is divided into 15 sectors of equal area: 6 red sectors, 5 blue, 3 yellow, and 1 green. In Problems, 7-14, consider the experiment of spinning the spinner once. Find the probability that the spinner lands on: Yellow A circular spinner is divided into 15 sectors of equal area: 6 red sectors, 5 blue, 3 yellow, and 1 green. In Problems, 7-14, consider the experiment of spinning the spinner once. Find the probability that the spinner lands on: Yellow or green.A circular spinner is divided into 15 sectors of equal area: 6 red sectors, 5 blue, 3 yellow, and 1 green. In Problems, 7-14, consider the experiment of spinning the spinner once. Find the probability that the spinner lands on: Red or blue A circular spinner is divided into 15 sectors of equal area: 6 red sectors, 5 blue, 3 yellow, and 1 green. In Problems, 7-14, consider the experiment of spinning the spinner once. Find the probability that the spinner lands on: Orange.A circular spinner is divided into 15 sectors of equal area: 6 red sectors, 5 blue, 3 yellow, and 1 green. In Problems, 7-14, consider the experiment of spinning the spinner once. Find the probability that the spinner lands on: Yellow, red, or green.A circular spinner is divided into 15 sectors of equal area: 6 red sectors, 5 blue, 3 yellow, and 1 green. In Problems, 7-14, consider the experiment of spinning the spinner once. Find the probability that the spinner lands on: Blue, red, yellow, or green.A circular spinner is divided into 15 sectors of equal area: 6 red sectors, 5 blue, 3 yellow, and 1 green. In Problems, 7-14, consider the experiment of spinning the spinner once. Find the probability that the spinner lands on: Purple Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing A club Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing A black card Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing A heart or diamond Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing A numbered card Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing The jack of clubs Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing An ace Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing A king or spade Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing A red queen Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing A black diamond Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of drawing 1 card from a standard 52 -card deck. In Problems 15-24, what is the probability of drawing A six or club In a family with 2 children, excluding multiple births, what is the probability of having 2 children of the opposite gender? Assume that a girl is as likely as a boy at each birth.In a family with 2 children, excluding multiple births, what is the probability of having 2 girls? Assume that a girl is as likely as a boy at each birth.A store carries four brands of DVD players: J,G,P , and S. From past records, the manager found that the relative frequency of brand choice among customers varied. Which of the following probability assignments for a particular customer choosing a particular brand of DVD player would have to be rejected? Why? APJ=.15,PG=.35,PP=.50,PS=.70 BPJ=.32,PG=.28,PP=.24,PS=.30 CPJ=.26,PG=.14,PP=.30,PS=.30 Using the probability assignments in Problem 27C. What is the probability that a random customer will not choose brand S ?Using the probability assignments in Problem 27C, what is the probability that a random customer will choose brand J or brand P ?Using the probability assignments in Problem 27C, what is the probability that a random customer will not choose brand J or brand P ?In a family with 3 children, excluding multiple births, what is the probability of having 2 boys and 1 girl, in that order? Assume that a boy is as likely as a girl at each birth.In a family with 3 children, excluding multiple births, what is the probability of having 2 boys and 1 girl, in any order? Assume that a boy is as likely as a girl at each birth.A keypad at the entrance of a building has 10 buttons labeled 0 through 9. What is the probability of a person correctly guessing a 4 -digit entry code?A keypad at the entrance of a building has 10 buttons labeled 0 through 9. What is the probability of a person correctly guessing a 4 -digit entry code if they know that no digits repeat?Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of dealing 5 cards from a standard 52 -card deck. In Problems 35-38, what is the probability of being dealt 5 black cards?Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of dealing 5 cards from a standard 52 -card deck. In Problems 35-38, what is the probability of being dealt 5 hearts?Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of dealing 5 cards from a standard 52 -card deck. In Problems 35-38, what is the probability of being dealt 5 face cards?Refer to the description of a standard deck of 52 cards and Figure 4 on page 384. An experiment consists of dealing 5 cards from a standard 52 -card deck. In Problems 35-38, what is the probability of being dealt 5 nonface cards?Twenty thousand students are enrolled at a state university. A student is selected at random, and his or her birthday (month and day, not year) is recorded. Describe an appropriate sample space for this experiment and assign acceptable probabilities to the simple events. What are your assumptions in making this assignment?In a three-way race for the U.S. Senate, polls indicate that the two leading candidates are running neck-and-neck, while the third candidate is receiving half the support of either of the others. Registered voters are chosen at random and asked which of the three will get their vote. Describe an appropriate sample space for this random survey experiment and assign acceptable probabilities to the simple events.Suppose that 5 thank-you notes are written and 5 envelopes are addressed. Accidentally, the notes are randomly inserted into the envelopes and mailed without checking the addresses. What is the probability that all the notes will be inserted into the correct envelopes?Suppose that 6 people check their coats in a checkroom. If all claim checks are lost and the 6 coats are randomly returned, what is the probability that all the people will get their own coats back?An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is 6.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is 8.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is less than 5.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is greater than 8.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is not 7or11.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is not 2,4or6.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is 1.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is 13.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is divisible by 3.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is divisible by 4.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is 7or11 (a “natural”).An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is 2,3or12 (“craps”).An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is divisible by 2or3.An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 (page 398 ) and, assuming each simple event is as likely as any other, find the probability of the sum of the dots indicated in Problems 43-56. Sum is divisible by 2 and 3.An experiment consists of tossing three fair (not weighted) coins, except that one of the three coins has a head on both sides. Compute the probability of obtaining the indicated results in Problems 57-62. 1 head An experiment consists of tossing three fair (not weighted) coins, except that one of the three coins has a head on both sides. Compute the probability of obtaining the indicated results in Problems 57-62. 2 heads An experiment consists of tossing three fair (not weighted) coins, except that one of the three coins has a head on both sides. Compute the probability of obtaining the indicated results in Problems 57-62. 3 heads An experiment consists of tossing three fair (not weighted) coins, except that one of the three coins has a head on both sides. Compute the probability of obtaining the indicated results in Problems 57-62. 0 heads An experiment consists of tossing three fair (not weighted) coins, except that one of the three coins has a head on both sides. Compute the probability of obtaining the indicated results in Problems 57-62. More than 1 head An experiment consists of tossing three fair (not weighted) coins, except that one of the three coins has a head on both sides. Compute the probability of obtaining the indicated results in Problems 57-62. More than 1 tail In Problems, 63-68, a sample space S is described. Would it be reasonable to make the equally likely assumption? Explain. A single card is drawn from a standard deck. We are interested in whether or not the card drawn is a heart, so an appropriate sample space is S=H,N.In Problems, 63-68, a sample space S is described. Would it be reasonable to make the equally likely assumption? Explain. A single fair coin is tossed. We are interested in whether the coin falls heads or tails, so an appropriate sample space is S=H,T.In Problems, 63-68, a sample space S is described. Would it be reasonable to make the equally likely assumption? Explain. A single fair die is rolled. We are interested in whether or not the number rolled is even or odd, so an appropriate sample space is S=E,O.In Problems, 63-68, a sample space S is described. Would it be reasonable to make the equally likely assumption? Explain. A nickel and dime are tossed. We are interested in the number of heads that appear, so an appropriate sample space is S=0,1,2.In Problems, 63-68, a sample space S is described. Would it be reasonable to make the equally likely assumption? Explain. A wheel of fortune has seven sectors of equal area colored red, orange, yellow, green, blue, indigo, and violet. We are interested in the color that the pointer indicates when the wheel stops, so an appropriate sample space is S=R,O,Y,G,B,I,V.In Problems, 63-68, a sample space S is described. Would it be reasonable to make the equally likely assumption? Explain. A wheel of fortune has seven sectors of equal area colored red, orange, yellow, red, orange, yellow, and red. We are interested in the color that the pointer indicates when the wheel stops, so an appropriate sample space is S=R,O,Y..(A) Is it possible to get 19 heads in 20 flips of a fair coin. Explain. (B) If you flipped a coin 40 times and got 37 heads, would you suspect that the coin was unfair? Why or why not? If you suspect an unfair coin, what empirical probabilities would you assign to the simple events of the sample space?(A) Is it possible to get 7 double 6s in 10 rolls of a pair of fair dice? Explain. (B) If you rolled a pair of dice 36 times and got 11 double 6s, would you suspect that the dice were unfair? Why or why not? If you suspect loaded dice, what empirical probability would you assign to the event of rolling a double 6 ?An experiment consists of rolling two fair (not weighted ) 4 -sided dice and adding the dots on the two sides facing up. Each die is numbered 1-4. Compute the probability of obtaining the indicated sums in Problems 71-78. 2 An experiment consists of rolling two fair (not weighted) 4 -sided dice and adding the dots on the two sides facing up. Each die is numbered 1-4. Compute the probability of obtaining the indicated sums in Problems 71-78. 3 An experiment consists of rolling two fair (not weighted) 4 -sided dice and adding the dots on the two sides facing up. Each die is numbered 1-4. Compute the probability of obtaining the indicated sums in Problems 71-78. 4 An experiment consists of rolling two fair (not weighted) 4 -sided dice and adding the dots on the two sides facing up. Each die is numbered 1-4. Compute the probability of obtaining the indicated sums in Problems 71-78. 5 An experiment consists of rolling two fair (not weighted) 4 -sided dice and adding the dots on the two sides facing up. Each die is numbered 1-4. Compute the probability of obtaining the indicated sums in Problems 71-78. 6 An experiment consists of rolling two fair (not weighted) 4 -sided dice and adding the dots on the two sides facing up. Each die is numbered 1-4. Compute the probability of obtaining the indicated sums in Problems 71-78. 7 An experiment consists of rolling two fair (not weighted) 4 -sided dice and adding the dots on the two sides facing up. Each die is numbered 1-4. Compute the probability of obtaining the indicated sums in Problems 71-78. An odd sum An experiment consists of rolling two fair (not weighted) 4 -sided dice and adding the dots on the two sides facing up. Each die is numbered 1-4. Compute the probability of obtaining the indicated sums in Problems 71-78. An even sum In Problems 79-86, find the probability of being dealt the given hand from a standard 52 -card deck. Refer to the description of a standard 52 -card deck on page 384. A5-card hand that consists entirely of red cards In Problems 79-86, find the probability of being dealt the given hand from a standard 52 -card deck. Refer to the description of a standard 52 -card deck on page 384. A5-card hand that consists entirely of face cards In Problems 79-86, find the probability of being dealt the given hand from a standard 52 -card deck. Refer to the description of a standard 52 -card deck on page 384. A6-card hand that contains exactly two face cards

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