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All Textbook Solutions for College Algebra
To get the best loan rates available, the Riches want to save enough money to place 20% down on a $160,000 home. They plan to make monthly deposits of $125 in an investment account that offers 8.5% annual interest compounded semiannually. Will the Riches have enough for a 20% down payment after five years of saving? How much money will they have saved?Karl has two years to save $10000 to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly?Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for 1 hour, and each successive day she will increase her study time by 30 minutes. How many hours will Keisha have studied after one week?A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell count be after 1 day?A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels 34 the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?Rachael deposits $1500 into a retirement fund each year. The fund earns 8.2% annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55? How much of that amount will be interest earned?A student is shopping for a new computer. He is deciding among 3 desktop computers and 4 laptop computers. What is the total number of computer options?A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are 3 types of breakfast sandwiches. 4 side dish options, and 5 beverage choices. Find the total number of possible breakfast specials.A family of five is having portraits taken. Use the Multiplication Principle to find how many ways the family can line up for the portrait.A family of five is having portraits taken. Use the Multiplication Principle to find how many ways the photographer can line up 3 of the family members.A family of five is having portraits taken. Use the Multiplication Principle to find how many ways the family can line up for the portrait if the parents are required to stand on each end.A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find how many ways the 7 actors can line up.A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find how many ways 5 of the 7 actors can be chosen to line up.An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?A sundae bar at a wedding has 6 toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?Find the number of rearrangements of the letters in the wordCARRIER.For the following exercises, assume that there are n ways an event A can happen, m ways an event B can happen, and that A and B are non-overlapping. 1. Use the Addition Principle of counting to explain how many ways event A or B can occur.For the following exercises, assume that there are n ways an event A can happen, m ways an event B can happen, and that A and B are non-overlapping. 2. Use the Multiplication Principle of counting to explain how many ways event A and B can occur.Answer the following questions. 3. When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use?Answer the following questions. 4. Describe how the permutation of ii objects differs from the permutation of choosing r objects from a set of n objects. Include how each is calculated.Answer the following questions. 5. What is the term for the arrangement that selects r objects from a set of ii objects when the order of the r objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement?For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 6. Let the set A=5,3,1,2,3,4,5,6 . How many ways are there to choose a negative or an even number from A?For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 7. Let the set B=23,16,7,2,20,36,48,72 s. How many ways are there to choose a positive or an odd number from A?For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 8. How many ways are there to pick a red ace or a club from a standard card playing deck?For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 9. How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow?For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 10. How many outcomes are possible from tossing a pair of coins?For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 11. How many outcomes are possible from tossing a coin and rolling a 6-sided die?For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 12. How many two-letter strings—the first letter from A and the second letter from B—can he formed from the sets A=b,cdandB=a,e,i,o,u?For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 13. How many ways are there to construct a string of 3 digits if numbers can be repeated?For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. 14. How many ways are there to construct a string of 3 digits if numbers cannot be repeated?For the following exercises, compute the value of the expression. 15. P(5,2)For the following exercises, compute the value of the expression. 16. P(8,4)For the following exercises, compute the value of the expression. 17. P(3,3)For the following exercises, compute the value of the expression. 18. P(9,6)For the following exercises, compute the value of the expression. 19. P(11,5)For the following exercises, compute the value of the expression. 20. C(8,5)For the following exercises, compute the value of the expression. 21. C(12,4)For the following exercises, compute the value of the expression. 22. C(26,3)For the following exercises, compute the value of the expression. 23. C(7,6)For the following exercises, compute the value of the expression. 24. C(10,3)For the following exercises, find the number of subsets in each given set. 25. {1,2,3,4,5,6,7,8,9,10}For the following exercises, find the number of subsets in each given set. 26. {a,b,c,...,z}For the following exercises, find the number of subsets in each given set. 27. A set containing 5 distinct numbers, letters, and 3 distinct symbols distinctFor the following exercises, find the number of subsets in each given set. 28. The set of even numbers from 2 to 28For the following exercises, find the number of subsets in each given set. 29. The set of two-digit numbers between 1 and 100 containing the digit 0For the following exercises, find the distinct number of arrangements. 30. The letters in the word “juggernaut”For the following exercises, find the distinct number of arrangements. 31. The letters in the word “academia”For the following exercises, find the distinct number of arrangements. 32. The letters in the word “academia” that begin and end in “a”For the following exercises, find the distinct number of arrangements. 33. The symbols in the string #,#,#,@,@,S,$,$,%,%%,%For the following exercises, find the distinct number of arrangements. 34. The symbols in the string #,#,#,@@,$,$,$,%,%,%that begin and end with % “The set, S consists of 900,000000 whole numbers, each being the same number of digits long. How many digits long is a number from S? (Hint: use the fact that a whole number cannot start with the digit 0.)The number of 5-element subsets from a set containing n elements is equal to the number of 6-element subsets from the same set. What is the value of n? (Hint: the order in which the elements for the subsets are chosen is not important.)Can C(n,r) ever equal P(n,r)? Explain.Suppose a set A has 2,048 subsets. How many distinct objects are contained in A?How many arrangements can be made from the letters of the word “mountains” if all the vowels must form a string?A family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in the front and 3 in the back a. How many arrangements are possible with no restrictions? b. How many arrangements are possible if the parents must sit in the front? C. How many arrangements are possible if the parents must be next to each other?A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?In horse racing, a “trifecta” occurs when a bettor wins by selecting the first three finishers in the exact order (1 St place, 2nd place, and 3rd place). How many different trifectas are possible if there are 14 horses in a race?A wholesale T-shirt company oilers sizes small, medium, large, and extra-large in organic or non- organic cotton and colors white, black, gray, blue, and red. How many different T-shirts are there to choose from?Hector wants to place billboard advertisements throughout the county for his new business. How many ways can Hector choose 15 neighborhoods to advertise in if there are 30 neighborhoods in the county?An art store has 4 brands of paint pens in 12 different colors and 3 types of ink. How many paint pens are there to choose from?How many ways can a committee of 3 freshmen and 4 juniors be formed from a group of 8 freshmen and 11 juniors?How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. 10 do this, he ranks the orchestra’s 10 cellists and 16 violinists in order of musical proficiency. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?A motorcycle shop has 10 choppers, 6 bobbers, and 5 café racers—different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 café racers for a weekend showcase?A skateboard shop stocks 10 types of board decks, 3 types of trucks, and 4 types of wheels. How many different skateboards can be constructed?Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to 11 customizable options?A car wash offers the following optional services to the basic wash: clear coat wax, triple foam polish, undercarriage wash, rust inhibitor, wheel brightener, air freshener, and interior shampoo. How many washes are possible if any number of options can be added to the basic wash?Susan bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?How many unique ways can a string of Christmas lights be arranged from 9 red, 10 green, 6 white, and 12 gold color bulbs?Find each binomial coefficient. (37) b. (411)Write in expanded form. a.(xy)5b.(2x+5y)3Find the sixth term of (3xy)9 without fully expanding the binomial.What is a binomial coefficient, and how it is calculated?What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?What is the Binomial Theorem and what is its use?When is it an advantage to use the Binomial Theorem? Explain.For the following exercises, evaluate the binomial coefficient. 5. (26)For the following exercises, evaluate the binomial coefficient. 6. (35)For the following exercises, evaluate the binomial coefficient. 7. (47)For the following exercises, evaluate the binomial coefficient. 8. (79)For the following exercises, evaluate the binomial coefficient. 9. (910)For the following exercises, evaluate the binomial coefficient. 10. (1125)For the following exercises, evaluate the binomial coefficient. 11. (617)For the following exercises, evaluate the binomial coefficient. 12. (199200)For the following exercises, use the Binomial Theorem to expand each binomial. 13. (4ab)3For the following exercises, use the Binomial Theorem to expand each binomial. 14. (5a+2)3For the following exercises, use the Binomial Theorem to expand each binomial. 15. (3a+2b)3For the following exercises, use the Binomial Theorem to expand each binomial. 16. (2x+3y)4For the following exercises, use the Binomial Theorem to expand each binomial. 17. (4x+2y)5For the following exercises, use the Binomial Theorem to expand each binomial. 18. (3x2y)4For the following exercises, use the Binomial Theorem to expand each binomial. 19. (4x3y)5For the following exercises, use the Binomial Theorem to expand each binomial. 20. (1x+3y)5For the following exercises, use the Binomial Theorem to expand each binomial. 21. (x1+2y1)4For the following exercises, use the Binomial Theorem to expand each binomial. 22. (xy)5For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. 23. (a+b)17For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. 24. (x1)18For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. 25. (a2b)15For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. 26. (x2y)8For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. 27. (3a+b)20For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. 28. (2a+4b)7For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. 29. (x3y)8For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 30. The fourth term of (2x3y)4For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 31. The fourth term of (3x2y)5For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 32. The third term of (6x3y)7For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 33. The eighth term of (7+5y)14For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 34. The seventh term of (a+b)11For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 35. The fifth term of (xy)7For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 36. The tenth term of (x1)12For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 37. The ninth term of (a3b2)11For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 38. The fourth term of (x312)10For the following exercises, find the indicated term of each binomial without fully expanding the binomial. 39. The eighth term of (y2+2x)For the following exercises, use the Binomial Theorem to expand the binomial f(x)=(x+3)4 . Then find and graph each indicated sum on one set of axes. 40. Find and graph f1(x) , such that f1(x) is the first term of the expansion.For the following exercises, use the Binomial Theorem to expand the binomial f(x)=(x+3)4 . Then find and graph each indicated sum on one set of axes. 41. Find and graph f2(x) , such that f2(x) is the sum of the first two terms of the expansion.For the following exercises, use the Binomial Theorem to expand the binomial f(x)=(x+3)4 . Then find and graph each indicated sum on one set of axes. 42. Find and graph f3(x) , such that f3(x) is the sum of the first three terms of the expansion.For the following exercises, use the Binomial Theorem to expand the binomial f(x)=(x+3)4 . Then find and graph each indicated sum on one set of axes. 43. Find and graph f4(x) , such that f4(x) is the sum of the first four terms of the expansion.For the following exercises, use the Binomial Theorem to expand the binomial f(x)=(x+3)4 . Then find and graph each indicated sum on one set of axes. 44. Find and graph f5(x) , such that f5(x) is the sum of the first five terms of the expansion.In the expansion of (5x+3y)n , each term has the form (nk)ankbk ,where k successively takes on the value 0,1,2....,n. If (nk)=(72) what is the corresponding term?In the expansion of (a+b)n, the coefficient of ankbk is the same as the coefficient of which other term?Consider the expansion of (x+b)40. What is the exponent of b in the kth term?Find (nk1)+(nk) and write the answer as a binomial coefficient in the form (nk) . Prove it. Hint:Use the fact that, for any integer p, such that p1 , p!=p(p1)! .Which expression cannot be expanded using the Binomial Theorem? Explain. a. (x22x+1) b. (a+4a5)8 c. (x3+2y2z)5 d. (3x22y3)12Construct a probability model for tossing a fair coin.A six-sided number cube is rolled. Find the probability of rolling a number greater than 2.A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.A card is drawn from a standard deck. Find the probability of drawing an ace or a king. Using the Complement Rule to Compute Probabilities We have discussed how to calculate the probability that an event will happen. Sometimes, we are Interested in finding the probability that an event will not happen. The complement of an event E, denoted E’,is the set of outcomes in the sample space that are not in E. For example, suppose we are interested in the probability that a horse will lose a race. If event W is the horse winning the race, then the complement of event W is the horse losing the race. To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1. P(E)=1P(E) The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is 19 ,the probability of the horse losing the race is simply. 119=89Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10.A child randomly selects 3 gumballs from a container holding 4 purple gumballs, 8 yellow gumballs, and 2 green gumballs. a. Find the probability that all 3 gumballs selected are purple. b. Find the probability that no yellow gumballs are selected. c. Find the probability that at least 1 yellow gumball is selected.What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain.What is a sample space?What is an experiment?What is the difference between events and outcomes? Give an example of both using the sample space of tossing a coin 50 times.The union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the union of two events from a probability model? How is it different?For the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated. Figure 3 6. Landing on redFor the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated. Figure 3 7. Landing on a vowelFor the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated. Figure 3 8. Not landing on blueFor the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated. Figure 3 9. Landing on purple or a vowelFor the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated. Figure 3 10. Landing on blue or a vowelFor the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated. Figure 3 11. Landing on green or blueFor the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated. Figure 3 12. Landing on yellow or a consonantFor the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated. Figure 3 13. Not landing on yellow or a consonantFor the following exercises, two coins are tossed. 14. What is the sample space?For the following exercises, two coins are tossed. 15. Find the probability of tossing two heads.For the following exercises, two coins are tossed. 16. Find the probability of tossing exactly one tail.For the following exercises, two coins are tossed. 17. Find the probability of tossing at least one tail.For the following exercises, four coins are tossed. 18. What is the sample space?For the following exercises, two coins are tossed. 19. Find the probability of tossing exactly two heads.For the following exercises, two coins are tossed. 20. Find the probability of tossing exactly three heads.For the following exercises, four coins are tossed. 21. Find the probability of tossing four heads or four tails.For the following exercises, four coins are tossed. 22. Find the probability of tossing all tails.For the following exercises, four coins are tossed. 23. Find the probability of tossing not all tails.For the following exercises, four coins are tossed. 24. Find the probability of tossing exactly two heads or at least two tails.For the following exercises, four coins are tossed. 25. Find the probability of tossing either two heads or three heads.For the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: 26. A clubFor the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: 27. A twoFor the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: 28. Six or sevenFor the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: 29. Red sixFor the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: 30. An ace or a diamondFor the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: 31. A non-aceFor the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: 32. A heart or a non-jackFor the following exercises, two dice are rolled, and the results are summed. 33. Construct a table showing the sample space of outcomes and sums.For the following exercises, two dice are rolled, and the results are summed. 34. Find the probability of rolling a sum of 3.For the following exercises, two dice are rolled, and the results are summed. 35. Find the probability of rolling at least one four or a sum of 8.For the following exercises, two dice are rolled, and the results are summed. 36. Find the probability of rolling an odd sum less than 9.For the following exercises, two dice are rolled, and the results are summed. 37. Find the probability of rolling a sum greater than or equal to 15.For the following exercises, two dice are rolled, and the results are summed. 38. Find the probability of rolling a sum less than 15.For the following exercises, two dice are rolled, and the results are summed. 39. Find the probability of rolling a sum less than 6 or greater than 9.For the following exercises, two dice are rolled, and the results are summed. 40. Find the probability of rolling a sum between 6 and 9, inclusive.For the following exercises, two dice are rolled, and the results are summed. 41.Find the probability of rolling a sum of 5 or 6.For the following exercises, two dice are rolled, and the results are summed. 42. Find the probability of rolling any sum other than 5 or 6.For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: 43. A head on the coin or a clubFor the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: 44. A tail on the coin or red aceFor the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: 45. A head on the coin or a face cardFor the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: No acesFor the following exercises, use this scenario: a bag of M&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M&Ms. Reaching into the bag, a person grabs 5 M&Ms. 47. What is the probability of getting all blue M&Ms?For the following exercises, use this scenario: a bag of M&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M&Ms. Reaching into the bag, a person grabs 5 M&Ms. 48. What is the probability of getting 4 blue M&Ms?For the following exercises, use this scenario: a bag of M&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M&Ms. Reaching into the bag, a person grabs 5 M&Ms. 49. What is the probability of getting 3 blue M&Ms?For the following exercises, use this scenario: a bag of M&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M&Ms. Reaching into the bag, a person grabs 5 M&Ms. 50. What is the probability of getting no brown M&Ms?Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80. After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80. A win occurs if the player has correctly selected 3,4, or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) 51. What is the percent chance that a player selects exactly 3 winning numbers?Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80. After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80. A win occurs if the player has correctly selected 3,4, or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) 52. What is the percent chance that a player selects exactly 4 winning numbers?Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80. After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80. A win occurs if the player has correctly selected 3,4, or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) 53. What is the percent chance that a player selects all 5 winning numbers?Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80. After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80. A win occurs if the player has correctly selected 3,4, or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) 54. What is the percent chance of winning?Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80. After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80. A win occurs if the player has correctly selected 3,4, or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) 55. How much less is a player’s chance of selecting 3 winning numbers than the chance of selecting either 4 or 5 winning numbers?Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).[34] 56. If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.)Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).[34] 57. If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).[34] 58. If you meet five U.S. citizens, what is the percent chance that three are elderly? (Round to the nearest tenth of a percent)Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).[34] 59. If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.)Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).[34] 60. It is predicted that by 2030, one in five U.S. citizens will be elderly. How much greater will the chances of meeting an elderly person be at that time? What policy changes do you foresee if these statistics hold true?Write the first four terms of the sequence defined by the recursive formula a1=2,an=an1+n.Evaluate 6!(53)!3!Write the first four terms of the sequence defined by the explicit formula a an=10n+3 .Write the first four terms of the sequence defined by the explicit formula an=n!n(n+1).Is the sequence 47,4721,8221,397,... arithmetic? If so, find the common difference.Is the sequence 2,4,8,16,... arithmetic? If so, find the common difference.An arithmetic sequence has the first term a1=18 and common difference d=8 . What are the first five terms?An arithmetic sequence has terms a3=11.7 and a8=14.6 . What is the first term?Write a recursive formula for the arithmetic sequence 20,10,0,10,...Write a recursive formula for the arithmetic sequence 0,12,1,32,... , and then find the 31stterm.Write an explicit formula for the arithmetic sequence 78,2924,3724,158,...How many terms are in the finite arithmetic sequence 12,20,28,...,172 ?Find the common ratio for the geometric sequence 2.5,5,10,20,...Is the sequence 4,16,28,40,... geometric? If so find the common ratio. If not, explain why.A geometric sequence has terms a7=16,384 and a9=262,144 . What are the first five terms?A geometric sequence has the first term a1=3 and common ratio r=12 What is the 8thterm?What are the first five terms of the geometric sequence a1=3,an=4an1Write a recursive formula for the geometric sequence 1,13,19,127,...Write an explicit formula for the geometric sequence 15,115,145,1135,...How many terms are in the finite geometric sequence 5,53,59,...,559,049Use summation notation to write the sum of terms 12m+5 from m=0 to m=5 .Use summation notation to write the sum that results from adding the number 13 twenty times.Use the formula for the sum of the first ii terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series 2.5,4,5.5,A ladder has 15 tapered rungs, the lengths of which increase by a common difference. The first rung is 5 inches long, and the last rung is 20 inches long. What is the sum of the lengths of the rungs?Use the formula for the sum of the first nterms of a geometric series to find S9 , for the series 12,6,3,32,...The fees for the first three years of a hunting club membership are given in Table 1. If fees continue to rise at the same rate, how much will the total cost be for the first ten years of membership? Year Membership Fees 1 $1,500 2 $1,950 3 $2,535 Table 1Find the sum of the infinite geometric series k145(13)k=1.A ball has a bounce-back ratio 35 . of the height of the previous bounce. Write a series representing the total distance traveled by the ball, assuming it was initially dropped from a height of 5 feet. What is the total distance? (Hint: the total distance the ball travels on each bounce is the sum of the heights of the rise and the fall.)Alejandro deposits $80 of his monthly earnings into an annuity that earns 6.25% annual interest, compounded monthly. How much money will he have saved after 5 years?The twins Sarah and Scott both opened retirement accounts on their 2lst birthday. Sarah deposits $4,800.00 each year. earning 5.5% annual interest, compounded monthly. Scott deposits $3,600.00 each year, earning 8.5% annual interest, compounded monthly. Which twin will earn the most interest by the time they are 55 years old? How much more?How many ways are there to choose a number from the set 10,6,4,10,12,18,24,32 that is divisible by either 4 or 6?In a group of 20 musicians, 12 play piano, 7 play trumpet, and 2 play both piano and trumpet. How many musicians play either piano or trumpet?How many ways are there to construct a 4-digit code if numbers can be repeated?A palette of water color paints has 3 shades of green, 3 shades of blue, 2 shades of red, 2 shades of yellow, and 1 shade of black. How many ways are there to choose one shade of each color?Calculate P(18,4) .In a group of 5 freshman, 10 sophomores, 3 juniors, and 2 seniors, how many ways can a president, vice president, and treasurer be elected?Calculate C(15,6)A coffee shop has 7 Guatemalan roasts, 4 Cuban roasts, and 10 Costa Rican roasts. How many ways can the shop choose 2 Guatemalan, 2 Cuban, and 3 Costa Rican roasts for a coffee tasting event?How many subsets does the set 1,3,5,...,99 have?A day spa charges a basic day rate that includes use of a sauna, pool, and showers. For an extra charge, guests can choose from the following additional services: massage, body scrub, manicure, pedicure, facial, and straight-razor shave. How many ways are there to order additional services at the day spa?How many distinct ways can the word DEADWOOD be arranged?How many distinct rearrangements of the letters of the word DEADWOOD are there if the arrangement must begin and end with the letter D?Evaluate the binomial coefficient (823) .Use the Binomial Theorem to expand (3x+12y)6Use the Binomial Theorem to write the first three terms of (2a+b)17.Find the fourth term of (3a22b)11 without fully expanding the binomial.For the following exercises, assume two die are rolled. 47. Construct a table showing the sample space.For the following exercises, assume two die are rolled. 48. What is the probability that a roll includes a 2?For the following exercises, assume two die are rolled. 49. What is the probability of rolling a pair?For the following exercises, assume two die are rolled. 50. What is the probability that a roll includes a 2 or results in a pair?For the following exercises, assume two die are rolled. 51.What is the probability that a roll doesn’t include a 2 or result in a pair?For the following exercises, assume two die are rolled. 52. What is the probability of rolling a 5 or a 6?For the following exercises, assume two die are rolled. 53. What is the probability that a roll includes neither a 5 nor a 6?For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.) 54. What is the percent chance that all the children attending the party prefer soda?For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.) 55. What is the percent chance that at least one of the children attending the party prefers milk?For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.) 56. What is the percent chance that exactly 3 of the children attending the party prefer soda?For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.) 57. What is the percent chance that exactly 3 of the children attending the party prefer milk?Write the first four terms of the sequence defined by the recursive formula a=14,an=2+an12Write the first four terms of the sequence defined by the explicit formula an=n2n1n! .Is the sequence 0.3,1.2,2.1,3,... arithmetic? If sofind the common difference.An arithmetic sequence has the first term a1=4 and common difference d=43 .What is the 6thterm?Write a recursive formula for the arithmetic sequence 2,72,5,132,... and then find the 22nd term.Write an explicit formula for the arithmetic sequence 15.6,15,14.4,13.8,... and then find the 32 term.Is the sequence 2,1,12,14,... . geometric? If sofind the common ratio. If not, explain why.What is the 11thterm of the geometric sequence 1.5,3,6,12,... ?Write a recursive formula for the geometric sequence 1,12,14,18,....Write an explicit formula for the geometric sequence 4,43,49,427,....Use summation notation to write the sum of terms 3k256k from k=3 to k=15 .A community baseball stadium has 10 seats in the first row, 13 seats in the second row, 16 seats in the third row, and so on. There are 56 rows in all. What is the seating capacity of the stadium?Use the formula for the sum of the first n terms of a geometric series to find k=170.2(5)k1 .Find the sum of the infinite geometric series. k=113(15)k1Rachael deposits $3,600 into a retirement fund each year. The fund earns 7.5% annual interest, compounded monthly. If she opened her account when she was 20 years old, how much will she have by the time she’s 55? How much of that amount was interest earned?In a competition of 50 professional ballroom dancers, 22 compete in the fox-trot competition, 18 compete in the tango competition. and 6compete in both the fox-trot and tango competitions. How many dancers compete in the foxtrot or tango competitions?A buyer of a new sedan can custom order the car by choosing from 5 different exterior colors, 3 different interior colors, 2 sound systems, 3 motor designs, and either manual or automatic transmission. How many choices does the buyer have?To allocate annual bonuses, a manager must choose his top four employees and rank them first to fourth. In how many ways can he create the ‘Top- Four” list out of the 32 employees?A rock group needs to choose 3 songs to play at the annual Battle of the Bands. How many ways can they choose their set if have 15 songs to pick from?A self-serve frozen yogurt shop has 8 candy toppings and 4 fruit toppings to choose from. How many ways are there to top a frozen yogurt?How many distinct ways can the word EVANESCENCE be arranged if the anagram must end with the letter E?Use the Binomial Theorem to expand (32x12y)5 .Find the seventh term of (x212)13 “without fully expanding the binomial.For the following exercises, use the spinner in Figure 1. Figure 1 24. Construct a probability model showing each possible outcome and its associated probability. (Use the first letter for colors.)For the following exercises, use the spinner in Figure 1. Figure 1 25. What is the probability of landing on an odd number?For the following exercises, use the spinner in Figure 1. Figure 1 26. What is the probability of landing on blue?For the following exercises, use the spinner in Figure 1. Figure 1 27. What is the probability of landing on blue or an odd number?For the following exercises, use the spinner in Figure 1. Figure 1 28. What is the probability of landing on anything other blue or an odd number?For the following exercises, use the spinner in Figure 1. Figure 1 29. A bowl of candy holds 16 peppermint, 14 butterscotch, and 10 strawberry flavored candies. Suppose a person grabs a handful of 7 candies. What is the percent chance that exactly 3 are butterscotch? (Show calculations and round to the nearest tenth of a percent.)