Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus Enhanced with Graphing Utilities
If n is a positive integer, show that ( n 0 )( n 1 )+( n 2 )+ ( 1 ) n ( n n )=0( 5 0 ) ( 1 4 ) 5 +( 5 1 ) ( 1 4 ) 4 ( 3 4 )+( 5 2 ) ( 1 4 ) 3 ( 3 4 ) 2 +( 5 3 ) ( 1 4 ) 2 ( 3 4 ) 3 +( 5 4 )( 1 4 ) ( 3 4 ) 4 +( 5 5 ) ( 3 4 ) 5 =?Stirling’s Formula An approximation for n! , when n is large, is given by n! 2n ( n e ) n ( 1+ 1 12n1 ) Calculate 12!,20!and25! on your calculator. Then use Stirling’s formula to approximate 12!,20!and25!1REIf n( A )=8 , n( B )=12 , and n( AB )=3 , find n( AB ) .3RE4RE5RE6RE7RE8RE9REIn Problems 10 and 11, compute the value of the given expression. P( 8,3 )In Problems 10 and 11, compute the value of the given expression. C( 8,3 )Stocking a Store A clothing store sells pure wool and polyester-wool suits. Each suit comes in 3 colors and 10 sizes. How many suits are required for a complete assortment?Baseball On a given day, the American Baseball League schedules 7 games. How many different outcomes are possible, assuming that each game is played to completion?Choosing Seats If 4 people enter a bus that has 9 vacant seats, in how many ways can they be sealed?Choosing a Team In how many ways can a squad of 4 relay runners be chosen from a track team of 8 runners?Baseball In how many ways can 2 teams from 14 teams in the American League be chosen without regard to which team is at home?Telephone Numbers Using the digits 0, 1, 2,...,9, how many 7-digit numbers can be formed if the first digit cannot be 0 or 9 and if the last digit is greater than or equal to 2 and less than or equal to 3? Repeated digits are allowed.18REBinary Codes Using the digits 0 and 1, how many different numbers consisting of 8 digits can be formed?Arranging Flags How many different vertical arrangements are there of 10 flags if 4 are white, 3 are blue, 2 are green, and 1 is red?Forming Committees A group of 9 people is going to be formed into committees of 4, 3, and 2 people. How many committees can be formed if: a. A person can serve on any number of committees? b. No person can serve on more than one committee?Birthday Problem For this problem, assume that a year has 365 days. a. In how many ways can 18 people have different birthdays? b. What is the probability that no 2 people in a group of 18 people have the same birthday? c. What is the probability that at least 2 people in a group of 18 people have the same birthday?Unemployment According to the U.S. Bureau of Labor Statistics, 6.2 of the U.S. labor force was unemployed in 2013. a. What is the probability that a randomly selected member of the U.S. labor force was unemployed in 2013? b. What is the probability that a randomly selected member of the U.S. labor force was not unemployed in 2013?24REEach of the numbers 1, 2,..., 100 is written on an index card, and the cards are shuffled. If a card is selected at random, what is the probability that the number on the card is divisible by 5? What is the probability that the card selected either is a 1 or names a prime number?At the Milex tune-up and brake repair shop, the manager has found that a car will require a tune-up with a probability of 0.6 , a brake job with a probability of 0.1 , and both with a probability of 0.02 . a. What is the probability that a car requires either a tune-up or a brake job? b. What is the probability that a car requires a tune-up but not a brake job? c. What is the probability that a car requires neither a tune-up nor a brake job?1CT2CT3CT4CT5CT6CT7CT8CT9CT10CT11CT12CT13CT14CT15CT16CT1CR2CR3CR4CR5CR6CR7CR8CR9CR10CR11CR12CR1AYU2AYUTrue or false The intersection of two sets is always a subset of their union. (pp. 2-3)4AYU5AYUIf the number of elements in a set is a nonnegative integer, we say that the set is ________.7AYUTrue or False If a task consists of a sequence of three choices in which there are p selections for the first choice, q selections for the second choice, and r selections for the third choice, then the task of making these selections can be done in p · q ·r different ways.9AYU10AYUIf n( A )=15 , n( B )=20 , and n( AB )=10 , find n( AB ) .If n( A )=30 , n( B )=40 , and n( AB )=45 , find n( AB ) .If n( AB )=50 , n( AB )=10 , and n( B )=20 , find n( A ) .If n( AB )=60 , n( AB )=40 , and n( A )=n( B ) , find n( A ) .In Problems 15-22, use ihe information given in the figure. How many are in set A ?In Problems 15-22, use ihe information given in the figure. How many are in set B ?In Problems 15-22, use ihe information given in the figure. How many are in set A or B ?In Problems 15-22, use ihe information given in the figure. How many are in set A or B ?In Problems 15-22, use ihe information given in the figure. How many are in set A or C ?In Problems 15-22, use ihe information given in the figure. How many are in A and B and C ?In Problems 15-22, use ihe information given in the figure. How many are in A and B and C ?In Problems 15-22, use ihe information given in the figure. How many are in A or B or C ?Shirts and Ties A man has 5 shirts and 3 ties. How many different shirt-and-tie arrangements can he wear?Blouses and Skirts A woman has 5 blouses and 8 skirts. How many different outfits can she wear?Four-digit Numbers How many four-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if the First digit cannot be 0? Repeated digits are allowed.Five-digit Numbers How many five-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if the first digit cannot be 0 or 1? Repeated digits are allowed.Analyzing Survey Data In a consumer survey of 500 people, 200 indicated that they would be buying a major appliance within the next month, 150 indicated that they would buy a car, and 25 said that they would purchase both a major appliance and a car. How many will purchase neither? How many will purchase only a car?Analyzing Survey Data In a student survey, 200 indicated that they would attend Summer Session I, and 150 indicated Summer Session II. If 75 students plan to attend both summer sessions, and 275 indicated that they would attend neither session, how many students participated in the survey?Analyzing Survey Data In a survey of 100 investors in the stock market, 50 owned shares in IBM 40 owned shares in AT&T 45 owned shares in GE 20 owned shares in both IBM and GE 15 owned shares in both AT&T and GE 20 owned shares in both IBM and AT&T a. How many of the investors surveyed did not have shares in any of the three companies? b. How many owned just IBM shares? c. How many owned just GE shares? d. I low many owned neither IBM nor GE? e. I low many owned either IBM or AT&T but no GE?30AYUDemographics The following data represent the marital status of males 18 years old and older in the U.S. in 2014. (a) Determine the number of males 18 years old and older who are widowed or divorced. (b) Determine the number of males 18 years old and older w ho are married, widowed, or divorced.32AYUStock Portfolios As a financial planner, you are asked to select one stock each from the following groups: 8 Dow Jones stocks, 15 NASDAQ stocks, and 4 global stocks. How many different portfolios are possible?Make up a problem different from any found in the text that requires the addition principle of counting to solve. Give it to a friend to solve and critique.35AYU0!= ; 1!= . (p. 642)True or False n!= ( n+1 )! n . (p. 642)A(n) __________ is an ordered arrangement of r objects chosen from n objects.A(n) ___________ is an arrangement of r objects chosen from n distinct objects, without repetition and without regard to order.P( n,r )= __________________.C( n,r )= _______________________.In Problems 7-14, find the value of each permutation. P( 6,2 )In Problems 7-14, find the value of each permutation. P( 7,2 )In Problems 7-14, find the value of each permutation. P( 4,4 )In Problems 7-14, find the value of each permutation. P( 8,8 )In Problems 7-14, find the value of each permutation. P( 7,0 )In Problems 7-14, find the value of each permutation. P( 9,0 )In Problems 7-14, find the value of each permutation. P( 8,4 )In Problems 7-14, find the value of each permutation. P( 8,3 )In Problems 15-22, use formula (2) to find the value of each combination. C( 8,2 )In Problems 15-22, use formula (2) to find the value of each combination. C( 8,6 )In Problems 15-22, use formula (2) to find the value of each combination. C( 7,4 )In Problems 15-22, use formula (2) to find the value of each combination. C( 6,2 )In Problems 15-22, use formula (2) to find the value of each combination. C( 15,15 )In Problems 15-22, use formula (2) to find the value of each combination. C( 18,1 )In Problems 15-22, use formula (2) to find the value of each combination. C( 26,13 )In Problems 15-22, use formula (2) to find the value of each combination. C( 18,9 )List all the ordered arrangements of 5 objects a , b , c , d , and e choosing 3 at a time without repetition. What is P( 5,3 ) ?List all the ordered arrangements of 5 objects a , b , c , d , and e choosing 2 at a time without repetition. What is P( 5,2 ) ?List all the ordered arrangements of 4 objects 1, 2, 3, and 4 choosing 3 at a time without repetition. What is P( 4,3 ) ?List all the ordered arrangements of 6 objects 1, 2, 3, 4, 5, and 6 choosing 3 at a time without repetition. What is P( 6,3 ) ?List all the combinations of 5 objects a , b , c , d , and e taken 3 at a time. What is C( 5,3 ) ?List all the combinationss of 5 objects a , b , c , d , and e taken 2 at a time. What is C( 5,2 ) ?List all the combinations of 4 objects 1, 2, 3, and 4 taken 3 at a time. What is C( 4,3 ) ?List all the combinationss of 6 objects 1, 2, 3, 4, 5, and 6 taken 3 at a time. What is C( 6,3 ) ?Forming Codes How many two-letter codes can be formed using the letters A , B , C , and D ? Repeated letters are allowed.Forming Codes How many two-letter codes can be formed using the letters A , B , C , D , and E ? Repeated letters are allowed.Forming Numbers How many three-digit numbers can be formed using the digits 0 and 1? Repeated digits are allowed.Forming Numbers How many three-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9? Repeated digits are allowed.Lining People Up In how many ways can 4 people be lined up?Stacking Boxes In how many ways can 5 different boxes be stacked?Forming Codes How many different three-letter codes are there if only the letters A , B , C , D , and E can be used and no letter can be used more than once?Forming Codes How many different four-letter codes are there if only the letters A , B , C , D , E , and F can be used and no letter can be used more than once?Stocks on the NYSE Companies whose stocks are listed on the New York Stock Exchange (NYSE) have their company name represented by 1, 2, or 3 letters (repetition of letters is allowed). What is the maximum number of companies that can be listed on the NYSE?Stocks on the NASDAQ Companies whose stocks are listed on the NASDAQ stock exchange have their company name represented by either 4 or 5 letters (repetition of letters is allowed). What is the maximum number of companies that can be listed on the NASDAQ?Establishing Committees In how many ways can a committee of 4 students be formed from a pool of 7 students?Establishing Committees In how many ways can a committee of 3 professors be formed from a department that has 8 professors?Possible Answers on a True/False Test How many arrangements of answers are possible for a true/false test with 10 questions?Possible Answers on a Multiple-choice Test How many arrangements of answers are possible in a multiple-choice test with 5 questions, each of which has 4 possible answers?Arranging Books Five different mathematics books are to be arranged on a students desk. How many arrangements are possible?Forming License Plate Numbers How many different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9, if: (a) Letters and digits may be repeated? (b) Letters may be repeated, but digits may not be repeated? (c) Neither letters nor digits may be repeated?Birthday Problem In how many ways can 2 people each have different birthdays? Assume that there are 365 days in a year.Birthday Problem In how many ways can 5 people all have different birthdays? Assume that there are 365 days in a year.Forming a Committee A student dance committee is to be formed consisting of 2 boys and 3 girls. If the membership is to be chosen from 4 boys and 8 girls, how many different committees are possible?Forming a Committee The student relations committee of a college consists of 2 administrators, 3 faculty members, and 5 students. Four administrators, 8 faculty members, and 20 students are eligible to serve. How many different committees are possible?Forming Words How many different 9-letter words (real or imaginary) can be formed from the letters in the word ECONOMICS?Forming Words How many different 11-letter words (real or imaginary) can be formed from the letters in the word MATHEMATICS?Selecting Objects An urn contains 7 white balls and 3 red balls. Three balls are selected. In how many ways can the 3 balls be drawn from the total of 10 balls: (a) If 2 balls are white and 1 is red? (b) If all 3 balls are white? (c) If all 3 balls are red?Selecting Objects An urn contains 15 red balls and 10 white balls. Five balls are selected. In how many ways can the 5 balls be drawn from the total of 25 balls: (a) If all 5 balls are red? (b) If 3 balls are red and 2 are white? (c) If at least 4 are red balls?Senate Committees The U.S. Senate has 100 members. Suppose that it is desired to place each senator on exactly 1 of 7 possible committees. The first committee has 22 members, the second has 13, the third has 10, the fourth has 5, the fifth has 16, and the sixth and seventh have 17 apiece. In how many ways can these committees be formed?Football Teams A defensive football squad consists of 25 players. Of these, 10 are linemen, 10 are linebackers, and 5 are safeties. How many different teams of 5 linemen, 3 linebackers, and 3 safeties can be formed?Baseball In the American Baseball League, a designated hitter may be used. How many batting orders is it possible for a manager to use? (There are 9 regular players on a team.)Baseball In the National Baseball League, the pitcher usually bats ninth. If this is the case, how many batting orders is it possible for a manager to use?Baseball Teams A baseball team has 15 members. Four of the players are pitchers, and the remaining 11 members can play any position. How many different teams of 9 players can be formed?World Series In the World Series the American League team (A) and the National League team (N) play until one team wins four games. If the sequence of winners is designated by letters (for example, NAAAA means that the National League team won the first game and the American League won the next four), how many different sequences are possible?Basketball Teams A basketball team has 6 players who play guard (2 of 5 starting positions). How many different teams are possible, assuming that the remaining 3 positions are filled and it is not possible to distinguish a left guard from a right guard?Basketball Teams On a basketball team of 12 players, 2 play only center, 3 play only guard, and the rest play forward (5 players on a team: 2 forwards, 2 guards, and 1 center). How many different teams are possible, assuming that it is not possible to distinguish a left guard from a right guard or a left forward from a right forward?Combination Locks A combination lock displays 50 numbers. To open it, you turn clockwise to the first number of the combination, then rotate counterclockwise to the second number, and then rotate clockwise to the third number. (a) How many different lock combinations are there? (b) Comment on the description of such a lock as a combination lock.Create a problem different from any found in the text that requires a permutation to solve. Give it to a friend to solve and critique.Create a problem different from any found in the text that requires a combination to solve. Give it to a friend to solve and critique.Explain the difference between a permutation and a combination. Give an example to illustrate your explanation.When the same probability is assigned to each outcome a sample space, the experiment is said to have _____________ outcomes.The _____________of an event E is the set of all outcomes in the sample space S that are not outcomes in the event E .True or False The probability of an event can never equal 0.True or False In a probability model, the sum of all probabilities is 1.In a probability model, which of the following numbers could be the probability of an outcome? 00.010.350.411.4In a probability model, which of the following numbers could be the probability of an outcome? 1.5 1 2 3 4 2 3 0 1 4Determine whether the following is a probability model.Determine whether the following is a probability model.Determine whether the following is a probability model.Determine whether the following is a probability model.In Problems 11-16, construct a probability model for each experiment. Tossing a fair coin twiceIn Problems 11-16, construct a probability model for each experiment. Tossing two fair coins onceIn Problems 11-16, construct a probability model for each experiment. Tossing two fair coins and then a fair dieIn Problems 11-16, construct a probability model for each experiment. Tossing a fair coin, a fair die, and then a fair coinIn Problems 11-16, construct a probability model for each experiment. Tossing three fair coins onceIn Problems 11-16, construct a probability model for each experiment. Tossing one fair coin three timesIn Problems 17-22, use the following spinners to construct a probability model for each experiment. Spin spinner I, then spinner II. What is the probability of getting a 2 or a 4, followed by Red?In Problems 17-22, use the following spinners to construct a probability model for each experiment. Spin spinner III, then spinner II. What is the probability of getting Forward, followed by Yellow or Green?In Problems 17-22, use the following spinners to construct a probability model for each experiment. Spin spinner I, then II, then III. What is the probability of getting a 1, followed by Red or Green, followed by Backward?In Problems 17-22, use the following spinners to construct a probability model for each experiment. Spin spinner II, then I, then III. What is the probability of getting Yellow, followed by a 2 or a 4, followed by Forward?In Problems 17-22, use the following spinners to construct a probability model for each experiment. Spin spinner I twice, then spinner II. What is the probability of getting a 2, followed by a 2 or a 4, followed by Red or Green?In Problems 17-22, use the following spinners to construct a probability model for each experiment. Spin spinner III, then spinner I twice. What is the probability of getting Forward, followed by a 1 or a 3, followed by a 2 or a 4?In Problems 23-26, consider the experiment of tossing a coin twice. The table lists six possible assignments of probabilities for this experiment. Using this table, answer the following questions. Which of the assignments of probabilities is(are) consistent with the definition of a probability model?In Problems 23-26, consider the experiment of tossing a coin twice. The table lists six possible assignments of probabilities for this experiment. Using this table, answer the following questions. Which of the assignments of probabilities should be used if the coin is known to be fair?25AYU26AYUAssigning Probabilities A coin is weighted so that heads is four times as likely as tails to occur. What probability should be assigned to heads? to tails?Assigning Probabilities A coin is weighted so that tails is twice as likely as heads to occur. What probability should be assigned to heads? to tails?Assigning Probabilities A die is weighted so that an odd-numbered face is twice as likely to occur as an even-numbered face. What probability should be assigned to each face?Assigning Probabilities A die is weighted so that a six cannot appear. All the other faces occur with the same probability. What probability should be assigned to each face?For Problems 31-34, the sample space is S={ 1,2,3,4,5,6,7,8,9,10 } Suppose that the outcomes are equally likely. Compute the probability of the event E={ 1,2,3 }For Problems 31-34, the sample space is S={ 1,2,3,4,5,6,7,8,9,10 } Suppose that the outcomes are equally likely. Compute the probability of the event F={ 3,5,9,10 }For Problems 31-34, the sample space is S={ 1,2,3,4,5,6,7,8,9,10 } Suppose that the outcomes are equally likely. Compute the probability of the event E : an even number.For Problems 31-34, the sample space is S={ 1,2,3,4,5,6,7,8,9,10 } Suppose that the outcomes are equally likely. Compute the probability of the event F: an odd number.For Problems 35 and 36, an urn contains 5 white marbles, 10 green marbles, 8 yellow marbles, and 7 black marbles. If one marble is selected, determine the probability that it is white.For Problems 35 and 36, an urn contains 5 white marbles, 10 green marbles, 8 yellow marbles, and 7 black marbles. If one marble is selected, determine the probability that it is black.In Problems 37-40, assume equally likely outcomes. Determine the probability of having 3 boys in a 3-child family.In Problems 37-40, assume equally likely outcomes. Determine the probability of having 3 girls in a 3-child family.In Problems 37-40, assume equally likely outcomes. Determine the probability of having 1 girl and 3 boys in a 4-child family.In Problems 37-40, assume equally likely outcomes. Determine the probability of having 2 girls and 2 boys in a 4-child family.For Problems 41-44, two fair dice are rolled. Determine the probability that the sum of the faces is 7.For Problems 41-44, two fair dice are rolled. Determine the probability that the sum of the faces is 11.For Problems 41-44, two fair dice are rolled. Determine the probability that the sum of the faces is 3.44AYUIn Problems 45-48, find the probability of the indicated event if P( A )=0.25 and P( B )=0.45 P( AB )ifP( AB )=0.15In Problems 45-48, find the probability of the indicated event if P( A )=0.25 and P( B )=0.45 P( AB )ifP( AB )=0.6In Problems 45-48, find the probability of the indicated event if P( A )=0.25 and P( B )=0.45 P( AB )ifA,BaremutuallyexclusiveIn Problems 45-48, find the probability of the indicated event if P( A )=0.25 and P( B )=0.45 P( AB )ifA,BaremutuallyexclusiveIf P( A )=0.60 , P( AB )=0.85 , and P( AB )=0.05 , find P( B ) .50AYU51AYU52AYU53AYUDoctorate Degrees According to the National Science Foundation, in 2013 there was a 17.0 probability that a doctoral degree awarded at a U.S. university was awarded55AYUFor Problems 57-60, a golf ball is selected at random from a container. If the container has 9 white balls, 8 green balls, and 3 orange balls, find the probability of each event.57AYU58AYU59AYU60AYUOn The Price Is Right, there is a game in which a bag is filled with 3 strike chips and 5 numbers. Lets say that the numbers in the bag are 0, 1, 3, 6, and 9. What is the probability of selecting a strike chip or the number 1?62AYU63AYU64AYU65AYU66AYU67AYUCheckout Lines Through observation, it has been determined that the probability for a given number of people waiting in line at the 5 items or less checkout register of a supermarket is as follows: Find the probability of: (a) At most 2 people in line (b) At least 2 people in line (c) At least 1 person in line69AYU70AYU71AYUBirthday Problem What is the probability that at least 2 people in a group of 35 people have the same birthday? Assume that there are 365 days in a year.73AYU1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE1CT2CT3CT4CT5CT6CT7CT8CT9CT10CT11CT12CT13CT14CT15CT16CT17CTGraph f( x )={ 3x2ifx2 3ifx=2 (pp.100-102)If f( x )={ xifx0 1ifx0 what is f( 0 ) ? (pp.100-102)The limit of a function f( x ) as x approaches c is denoted by the symbol _______. (a) lim xc f( x ) (b) lim fc f( x ) (c) lim xc f( x ) (d) lim cf f( x )If a function f has no limit as x approaches c , then we say that lim xc f( x ) ______.True or False lim xc f( x )=N may be described by saving that the value of f( x ) gets closer to N as x gets closer to c but remains unequal to c .True or False lim xc f( x ) exists and equals some number for any function function f as long as c is in the domain of f .lim x2 ( 4 x 3 )lim x3 ( 2 x 2 +1 )