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CHAPTER 9 REVIEW COUNTING PRINCIPLE AND PROBABILITY A. Consider the club: N = {Arnold, Bob, Cindy, Don, Ellen}. List and count the different ways of electing each of the following slates of officers. 1) A president and a secretary if one person can hold both offices. 2) A president, a secretary and a treasurer, if the president must be a man and the other two must be women. (assume that no one can hold more than one office) 3) List the possible ways club N could appoint a committee of three members. (assume that no one can hold more than one office) B. Construct a tree diagram and list all the possible results when three fair coins are tossed. 1) At least two heads 2) More than two heads 3) No more than two heads 4) Two tails C. Construct a tree diagram and list all the possible results of making a salad. The Salad-To-Go Deli offers a variety of salads. There are 3 choices for lettuce: iceberg, romaine, or Boston. There are 4 choices for toppings: tomatoes, cucumbers, bacon bits, or croutons. There are 3 choices for dressing: Italian, Russian or Blue cheese. (you can only make a salad with one choice from each category. i.e. you cannot make a salad with romaine, tomatoes, croutons and Italian dressing.) 1) How many different types of salads could be offered? 2) How many types of salads could you make if you only want to use iceberg lettuce? 3) How many types of salads could you make if you want cucumbers and Italian dressing? 4) How many types of salads could you make if the Deli had no more bacon bits or romaine lettuce? D. For the following problems use the counting principle. 1) Sam doesn’t know any of the answers for a twelve-question true or false test, so he must guess on all the questions. How many different ways can he mark his answer sheet? 2) 3) 4) How many different seven-digit phone numbers are there if the first two digits cannot be ones or zeros? How many different results are possible if five coins are tossed? How many different license “numbers” can a state issue if each includes two letters followed four digits and the first and last digits cannot be zeros? 5) How many three-digit numbers can be written using the digits 0, 1, 2, 3, 4, 5? (repetitions allowed , but the first digit cannot be 0) by are 6) Andy, Betty, Clyde, Dawn, Evan and Felicia have reserved six seats in a row at the theatre starting at an aisle seat. In how many ways can they arrange themselves? 7) How many different three-digit even numbers can be written using digits from the set {1, 2, 3, 4, 5}? 8) There are 7 members on the Board of Directors of the Ross Chemical Corp. as follows: Alice Johnson, Steve Bachtel, George Klagman, Jeremy Billingsley, Shawn Washington, Wallace McKenzie and Colleen Brooks. At any stockholder meeting these members are seated at a straight head table. a) In how many different ways can the Board members be seated? b) Jeremy and Wallace are bitter enemies and cannot be seated together. Therefore, they are seated in the last seat on each end. How many different seating arrangements are possible? c) How many number of ways can the Board members be seated if the two women must be seated next to each other? E) In a black velvet bag there are 12 red marbles, 20 green marbles, 4 orange marbles and 8 blue marbles. 1) Find the probability that you will pick a green marble. 2) Find the probability that you will pick a blue or red marble. 3) Find the probability that you will not pick an orange marble. 4) Find the probability that you will not pick a green or red marble. F. A single card is drawn from an ordinary 52-card deck. Find the probability of drawing the following: a) seven of hearts b) spades c) a diamond or a jack d) not an ace e) a number greater than 8(aces low) f) a black jack or a six of hearts g) a multiple of 2 or a heart h) a five or an odd number less than seven (aces low) G. Two fair dice are rolled. Find the probability that the sums are the following: a) a five b) an even number c) a seven or a two d) a multiple of three or an even number e) a number greater than four f) a double or a number greater than 9 H. The following chart represents the number of athletes declaring their major. Education Accounting Computer programming Communication Total Baseball 24 Soccer 13 Basketball 19 Total 56 30 25 18 73 16 21 91 19 22 79 15 25 77 50 68 247 Based on the above data, find the probability of each of the following events. a) the athlete was a baseball player b) the declared major was education c) the athlete was a soccer or basketball player d) a soccer player’s declared major is communications ANSWER KEY: A. 1) president, secretary and the letters are the initials of each person: 25ways AA, AB, AC, AD, AE, BA, BB, BC, BD, BE, CA, CB, CC, CD, CE, DA, DB, DC, DD, DE, EA, EB, EC, ED, EE 2) president, secretary, treasurer 6 ways: BCE, BEC, ACE, AEC, DCE, DEC 3) 10 ways: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE B. 1) HHH, HHT, HTH, THH 3) TTT, HHT, HTH, THH, TTH, THT, HTT C. 1) 36 salads 2) 12 salads 2) HHH 4) TTH, THT, HTT 3) 3 salads 4) 18 salads D. 1) 2 12 = 4096 2) 8(8)(10)(10)(10)(10)(10) = 6400000 3) 2 5 = 32 4) 26(26)(9)(10)(10)(9) = 5475600 5) 5(6)(6) = 180 6) 6(5)(4)(3)(2)(1) = 720 7) 5(5)(2) = 50 8) a) 7∙6∙5∙4∙3∙2∙1 = 5040 b) 2∙ 5∙ 4∙ 3∙ 2∙ 1∙ 1 = 240 c) 2∙ 1∙ 5∙ 4∙ 3∙ 2∙ 1 = 240 then 240(6) = 1440 E. 1) 20/44 = 5/11 2) 8/44 + 12/44 = 5/11 3) 40/44 = 10/11 4) 12/44 = 3/11 F. a) 1/52 b) 13/52 = ¼ c) (13/52 + 4/52) – 1/52 = 4/13 d) 48/52 = 12/13 e) 8/52 = 2/13 f) 2/52 + 1/52 = 3/52 g) (20/52 + 13/52) – 5/52 = 7/13 h) (4/52 + 12/52) – 4/52 = 3/13 G. a) 4/36 = 1/9 b) 18/36 = ½ c) 7/36 d) (12/36 + 18/36) – 6/36 = 2/3 e) 30/36 = 5 /6 f) (6/36 + 6/36) – 2/36 = 5/18 H. a) 91/247 = 7/19 b) 56/247 c) 79/247 + 77/247 = 12/19 d) 22/247