Exam 2 Review (Sections Covered: 3.1, 3.3, 6.1-6.4, 7.1) 1. Write a system of linear inequalities that describes the shaded region. 5x + 2y 30 x + 2y 12 x 0 y 0 2. Write a system of linear inequalities that describes the solution set. 7x + 6y 86 x + y 10 x 0 y 5
3. Solve the linear programming problem. Maximize P = 3x + 5y subject to 2x + y 16 2x + 3y 24 x 0 y 0 (a) Find the corner points of the solution set. (b) Find the maximum. 2 Fall 2016, Maya Johnson
4. Solve the linear programming problem. Minimize C = 2x + 4y subject to 4x + 2y 40 2x + 3y 30 x 0 y 0 (a) Find the corner points of the solution set. (b) Find the minimum. 3 Fall 2016, Maya Johnson
5. Let the universal set U = {u, v, w, x, y, z} with sets A = {u,v,y,z}, B = {x,y,z}, and C = {w, x}. Determine whether the following statements are true or false. (a) x, y B (b) {x, y, z} B (c) {u, w} / A (d) {u, y} A 6. Let the universal set U = { 3, 2, 1, 0, 1, 2, 3} with sets A = { -2,0,2}, B = { -3,-1,1,3}, and C = { 2, 1, 3}. Determine whether the following statements are true or false. (a) A has 20 subsets. (b) B c = (c) (A B) c = U (d) (A C) = { 2, 1, 0, 2, 3} (e) C (f) A and B are disjoint sets. 7. Let U be a universal set with sets A and B. Determine whether the following statements are true or false. (a) (U) c = (b) ( ) c = U (c) (A B) c = A c B c (d) B = B (e) B (f) A = 4 Fall 2016, Maya Johnson
8. Write venn diagrams to represent each of the following sets. (a) A B C c (b) A c B C (c) (A B) C (d) (A B c ) C 5 Fall 2016, Maya Johnson
9. Let U = {-9, -6, -1, 2, 5, 7, 11, 13, 17, 19}, A = {-9, -1, 5, 11, 17}, B = {-6, 2, 7, 13, 19}, and C = {-9, -6, 2, 5, 13, 17}. Find each set using roster notation. (a) (A B) C (b) (A B C) c (c) (A B C) c 10. Let U = { 8, 4, 2, 1, 3, 6, 9, 12, 16, 18}, A = { 8, 2, 3, 9, 16}, B = { 4, 1, 6, 12, 18}, and C = { 8, 4, 1, 3, 12, 16}. List the elements of each set. (a) A c (B C c ). (b) (A B c ) (B C c ) (c) (A B) c C c 6 Fall 2016, Maya Johnson
11. If n(b) = 13, n(a B) = 24, and n(a B) = 6, find n(a). 12. In a survey of 400 people, a pet food manufacturer found that 250 owned a bird, 150 owned a snake, and 75 owned neither a bird or a snake. (a) How many owned a bird or a snake? (b) How many owned both a bird and a snake? 13. In a survey of 300 members of a local sports club, 180 members indicated that they plan to attend the next Summer or Winter Olympic Games, 150 members indicated that they plan to attend the next Summer Olympic Games, and 90 indicated that they plan to attend the next Winter Olympic Games. How many members of the club plan to attend (a) Both of the games? (b) Exactly one of the games? (c) The Summer Olympic Games only? (d) None of the games? 7 Fall 2016, Maya Johnson
14. Let A and B be subsets of a universal set U and suppose n(u) = 48, n(a) = 13, n(b) = 23, and n(a B) = 8. Compute: (a) n(a c B) (b) n(b c ) (c) n(a c B c ) (d) How many subsets does B have? (e) How many proper subsets does B have? 15. Let A, B, and C be sets in a universal set U. We are given n(u) = 66, n(a) = 32, n(b) = 33, n(c) = 33, n(a B) = 16, n(a C) = 10, n(b C) = 18, n(a B C c ) = 9. Find the following values. (a) n((a B C) c ) (b) n((a c B c ) C) (c) n((a c B c ) C) 8 Fall 2016, Maya Johnson
16. Use the following information to determine the number of people in each region of the Venn Diagram. A group of 295 students were asked which of these sports they participated in during high school. 44 students participated in all of these sports. 87 students participated in basketball and track. 39 students participated in basketball and tennis but not track. 79 students participated in track but not tennis. 155 students participated in basketball. 142 students did not participate in tennis. 103 students participated in exactly one sport. a = b = Tennis a b Track c c = e d = d f e = g f = h Basketball g = h = 9 Fall 2016, Maya Johnson
17. Use the following information to determine the number of people in each region of the Venn Diagram. 251 people were asked which of these instruments that they could play: Piano, Drums, or Guitar. 20 people could play none of these instruments. 34 people could play all three of these instruments. 79 people could play drums or guitar but could not play piano. 115 people could play guitar. 130 people could play at least two of these instruments. 28 people could play piano and guitar but could not play drums. 78 people could play piano and drums. a = Guitar Piano b = b c = a e c d = d f e = g f = h Drums g = h = 10 Fall 2016, Maya Johnson
18. A group of students were asked which of these sports they play. The information was recorded in the Venn Diagram. Use the the Venn Diagram to answer these questions. Let T = Tennis, F = Football, and B = Basketball. a = 43 Football Tennis b = 38 a b e c c = 45 d = 8 d f e = 22 g f = 16 h Basketball g = 12 h = 35 (a) How many students play Football or Basketball but not Tennis? (b) How many students do not play Football? (c) How many students play tennis or do not play basketball? n(t c (B F )) (d) n(t (B F c )) 11 Fall 2016, Maya Johnson
19. In recent years, a state has issued license plates using a combination of three digits, followed by three letters of the alphabet followed by another three digits. How many different license plates can be issued using this configuration? 20. Complete the following. (a) How many seven-digit telephone numbers are possible if the first digit must be nonzero? (b) How many international direct-dialing numbers for calls within the United States and Canada are possible if each number consists of a 1 plus a three-digit area code (the first digit of which must be nonzero) and a number of the type described in part (a)? 21. A state makes license plates with three letters followed by four digits. (a) How many license plates are possible? (b) If no repetition of the letters is permitted, how many different license plates are possible? (c) If no repetition of letters or digits is permitted, how many different license plates are possible? (d) How many license plates have no repetition of letters or digits and begin with a vowel? 12 Fall 2016, Maya Johnson
22. A company car that has a seating capacity of eight is to be used by eight employees who have formed a car pool. If only three of these employees can drive, how many possible seating arrangements are there for the group? 23. There are four families attending a concert together. Each family consists of 1 male and 2 females. In how many ways can they be seated in a row of twelve seats if (a) There are no restrictions? (b) Each family is seated together? (c) The members of each gender are seated together? 24. Alex, Mark, Sue, Bill, and Maggie attend the movie theater. Assume that Sue and Maggie are female and that Alex, Mark, and Bill are male. How many ways can they be seated if (a) There are no restrictions? (b) The females sit together and the males sit together? (c) Mark and Sue want to sit together? 13 Fall 2016, Maya Johnson
25. At a college library exhibition of faculty publications, two mathematics books, four social science books, and three biology books will be displayed on a shelf. (Assume that none of the books are alike.) (a) In how many ways can the nine books be arranged on the shelf? (b) In how many ways can the nine books be arranged on the shelf if books on the same subject matter are placed together? 26. Find the number of distinguishable arrangements of each of the following words. (a) acdbens (b) baaaben (c) aaabbba 27. In how many ways can a subcommittee of six be chosen from a Senate committee of six Democrats and five Republicans if (a) All members are eligible? (b) The subcommittee must consist of three Republicans and three Democrats? 14 Fall 2016, Maya Johnson
28. In how many different ways can a panel of 12 jurors and 2 alternates be chosen from a group of 16 prospective jurors? 29. From a shipment of 25 transistors, 6 of which are defective, a sample of 9 transistors is selected at random. (a) In how many different ways can the sample be selected? (b) How many samples contain exactly 3 defective transistors? (c) How many samples contain no defective transistors? (d) How many samples contain at least 5 defective transistors? 15 Fall 2016, Maya Johnson
30. A box contains 8 red marbles, 8 green marbles, and 10 black marbles. A sample of 12 marbles is to be picked from the box. (a) How many samples contain at least 1 red marble? (b) How many samples contain exactly 4 red marbles and exactly 3 black marbles? (c) How many samples contain exactly 7 red marbles or exactly 6 green marbles? (d) How many samples contain exactly 5 green marbles or exactly 3 black marbles? 31. Suppose we have 20 people on a committee. How many subcommittees contain one president, one vice president and six cabinet members? 32. Ten runners are competing in a half-marathon. How many ways can we award one 1st place prize, one 2nd place prize, one 3rd place prize, and four 4th place prizes? 16 Fall 2016, Maya Johnson
33. Consider the sample space S = {s, t, n}. How many total events are there for this sample space? 34. Let S = {5, 9, 12} be a sample space associated with an experiment. (a) List all events of this experiment. (b) How many events of S contain the number 5? (c) How many events of S contain the number 12 or the number 5? 35. An experiment consists of tossing a coin and observing the side that lands up and then rolling a fair 4-sided die and observing the number rolled. Let H and T represent heads and tails respectively. (a) Describe the sample space S corresponding to this experiment. (b) What is the event E 1 that an even number is rolled? (c) What is the event E 2 that a head is tossed or a 3 is rolled? (d) What is the event E 3 that a tail is tossed and an odd number is rolled? 17 Fall 2016, Maya Johnson
36. The numbers 3, 4, 5, and 7 are written on separate pieces of paper and put into a hat. Two pieces of paper are drawn at the same time and the product of the numbers is recorded. Find the sample space. 37. A jar contains 8 marbles numbered 1 through 8. An experiment consists of randomly selecting a marble from the jar, observing the number drawn, and then randomly selecting a card from a standard deck and observing the suit of the card (hearts, diamonds, clubs, or spades). (a) How many outcomes are in the sample space for this experiment? (b) How many outcomes are in the event an even number is drawn? (c) How many outcomes are in the event a number more than 1 is drawn and a red card is drawn? (d) How many outcomes are in the event a number less than 2 is drawn or a club is not drawn? 18 Fall 2016, Maya Johnson