Probability, Combinatorics, and Overlapping Sets For questions in the Quantitative Comparison format (“Quantity A” and “Quantity B” given), the answer choices are always as follows: (A) Quantity A is greater. (B) Quantity B is greater. (C) The two quantities are equal (D) The relationship cannot be determined from the information given, ‘Where answer choices do not appear on Quantitative Comparison questions in this book, you should choose A, B, C or D based on the above. For questions followed by a numeri enty box CL} you ate to enter your own answer in L_] the box. For questions followed by a fraction-style numeric entry box =, you are to enter your answer in the form of a fraction. You are not required to reduce fractions. For example, if or any equivalent fraction. the answer is —, you may enter 100 All numbers used are real numbers. All figures are assumed to lie in a plane unless otherwise indicated. Geometric figures are not necessarily drawn to scale, You should assume, however, that lines that appear to be straight are actually straight, points on a line are in the order shown, and all geometric objects are in the relative positions shown. Coordinate systems, such as xy-planes and number lines, as well as graphical data presentations, such as bar charts, circle graphs, and line graphs, are drawn to scale. A symbol that appears more than once in a question has the same ‘meaning throughout the question. 1. A number is randomly chosen from a list of 10 consecutive positive integers. What is the probability that the number selected is greater than the average (arithmetic mean) of all 10 integers? 3 (A) — 1 2 (B) = 51 (C) — 2 o 10 w+ 5 2. A number is randomly chosen from the first 100 positive integers. What is the probability that it is a multiple of 3? 32 a) 100 33 py 2 ®) 100 1 © = 3 34 (D) — 100 (E) & 33. A restaurant menu has several options for tacos. There are 3 types of shells, 4 types of meat, 3 types of cheese, and 5 types of salsa. How many distinct tacos can be ordered assuming that any order contains exactly one of each of the above choices? [ A history exam features five questions. Three of the questions are multiple-choice with four options each. The other two questions are true or false. If Caroline selects one answer for every question, how many different ways can she answer the exam? [ 1 . The probability is a that a certain coin will turn up heads on any given - wa i toss and the probability is — that a number cube with faces numbered 1 to 6 will turn up any particular number. What is the probability of turning up a heads and a 6? 1 (A) = 36 (B) (©) (D) Ble aloagl|—oe () — 3 6. An integer is randomly chosen from 2 to 20 inclusive. What is the probability that the number is prime? Give your answer as a fraction. i 7. An Italian restaurant boasts 320 distinct pasta dishes. Each dish contains exactly 1 pasta, 1 meat, and 1 sauce. If there are 8 pastas and 4 meats available, how many sauces are there to choose from? [8. A 10-student class is to choose a president, vice president, and secretary from the group. If no person can occupy more than one post, in how many ways can this be accomplished? [| eo . BurgerTown offers many options for customizing a burger. There are 3 types of meats and 7 condiments: lettuce, tomatoes, pickles, onions, ketchup, mustard, and special sauce. A burger must include meat, but may include as many or as few condiments as the customer wants. How many different burgers are possible? (A) (B) (©) (D) () 8! (3)(7!) (3)(8!) (8)2”) @)@’) it 10. The probability of rain is @ for any given day next week. What is the probability that it will rain on both Monday and Tuesday? (A) (B) 3 J, 36 wlaalag|a@ 2 3 11. How many five-digit numbers can be formed using the digits 5, 6, 7, 8, 9, 0 if no digits can be repeated? (A) 64 (B) 120 (C) 240 (D) 600 (E) 72012. A bag contains 3 red, 2 blue, and 7 white marbles. If a marble is randomly chosen from the bag, what is the probability that it is not blue? Give your answer as a fraction. 13. A man has 3 different suits, 4 different shirts, 2 different pairs of socks, and 5 different pairs of shoes. If an outfit consists of exactly 1 suit, 1 shirt, 1 pair of socks, and 1 pair of shoes, how many different outfits can be made with the man’s clothing? [| A state issues automobile license plates that begin with two letters selected from a 26-letter alphabet, followed by four numerals selected from the digits 0 through 9, inclusive. Repeats are permitted. For example, one possible license plate combination is GF3352. Quantity A The number of possible unique Quantity B 6,000,000 14. license plate combinations 15. A bag contains 6 black chips numbered 1-6 respectively and 6 white chips numbered 1-6 respectively. If Pavel reaches into the bag of 12 chips and removes 2 chips, one after the other, without replacing them, what is the probability that he will pick black chip #3 and then white chip #3? Give your answer as a fraction.ilTarik has a pile of 6 green chips numbered 1 through 6 respectively and another pile of 6 blue chips numbered 1 through 6 respectively. Tarik will randomly pick 1 chip from the green pile and 1 chip from the blue pile. uantity A Quantity B The probability that both chips 1 selected by Tarik will display a = 16. number less than 4 2 17. A bag contains 6 red chips numbered 1 through 6, respectively, and 6 blue chips numbered 1 through 6, respectively. If 2 chips are to be picked sequentially from the bag of 12 chips, without replacement, what is the probability of picking a red chip and then a blue chip with the same number? Give your answer as a fraction. CL] LC] Ina school of 150 students, 75 study Latin, 110 study Spanish, and 11 study neither. Quantity A . Quantity B The number of students who study 4G 18. only Latin 19. How many 10-digit numbers can be formed using only the digits 2 and 5? (A) 219 (B) (226!)© G6) 10! 2 (E) 10! (D)20. A 6-sided cube has faces numbered 1 through 6. If the cube is rolled twice, what is the probability that the sum of the two rolls is 8? 1 @) = 9 w + 8 5 © 2 36 1 ©) = 6 © 2 36 21. A certain coin with heads on one side and tails on the other has a A, probability of landing on heads. If the coin is flipped 5 times, how many distinct outcomes are possible if the last flip must be heads? Outcomes are distinct if they do not contain exactly the same results in exactly the same order. [_ Ina class of 25 students, each student studies either Spanish, Latin, or French, or two of the three, but no students study all three languages. 9 study Spanish, 7 study Latin, and 5 study exactly two languages. Quantity A The number of students who study Quantity B22. French 14 23. Pedro has a number cube with 24 faces and an integer between 1 and 24 on each face. Every number is featured exactly once. When he rolls, what is the probability that the number showing is a factor of 24? Give your answer as a fraction. CL] CL]24. A baby has x total toys. If 9 of the toys are stuffed animals, 7 of the toys were given to the baby by her grandmother, 5 of the toys are stuffed animals given to the baby by her grandmother, and 6 of the toys are neither stuffed animals nor given to the baby by her grandmother, what is the value of x? [ 25. How many integers between 2,000 and 3,999 have a ones digit that is a prime number? [| 26. A group of 12 people who have never met are in a classroom. How many handshakes are exchanged if each person shakes hands exactly once with each of the other people in the room? (A) 12 27. A class consists of 12 girls and 20 boys. One quarter of the girls in the class have blue eyes. If a child is selected at random from the class, what is the probability that the child is a girl who does not have blue eyes? 3 ay 2 ws @) 32© 2B wo) = 32 . 29 © = 321 28. A certain coin with heads on one side and tails on the other has a — probability of landing on heads. If the coin is flipped 3 times, what is the probability of flipping 2 tails and 1 head, in any order? a + 8 w 1 3 © 2 “8 o 2 8 (E) A 3 29. A number cube has six faces numbered 1 through 6. If the cube is rolled twice, what is the probability that at least one of the rolls will result in a number greater than 4? a 2 9 1 ® — 3 @ 4 9 w 2 9@ 2 3 30. 100 tiles are labeled with the integers from 1 to 100 inclusive; no numbers are repeated. If Alma chooses 1 tile at random, replaces it in the group, and chooses another tile at random, what is the probability that the product of the two integer values on the tiles is odd? (A) 2 8 1 (B) a (©) A 3 1 @) — 2 ©) ss 431. If the word “WOW?” can be rearranged in exactly 3 ways (WOW, OWW, WWO), how many different arrangements of the letters in “MISSISSIPPI” are possible? [ 1 The probability of rain is — on any given day next week. 2 Quantity A Quantity B The probability that it rains on at 127 32. least one of the 7 days next week 128 33. Two number cubes with six faces numbered with the integers from 1 through 6 are tossed, What is the probability that the sum of the exposed faces on the cubes is a prime number? Give your answer as a fraction. 34, Jan and 5 other children are in a classroom. The principal of the school will choose 2 of the children at random. What is the probability that Jan will be chosen? (A) (B) wloaunls2 O Fz 5 7 (@) — 15 () i 2 35. The probability that Maria will eat breakfast on any given day is 0.5. The probability that Maria will wear a sweater on any given day is 0.3. The two probabilities are independent of each other. Quantity A i Quantity B The probability that Maria eats 08 breakfast or wears a sweaterThe probability of rain in Greg’s town on Tuesday is 0.3. The probability that Greg’s teacher will give him a pop quiz on Tuesday is 0.2. The events occur independently of each other. Quantity A Quantity B The probability that either or both The probability that neither event 36. events occur occurs 1 37. A certain city has a e chance of rain occurring on any given day. In any given 3-day period, what is the probability that the city experiences rain? (A) (B) (©) (D) () BENI= N N[SwinY|owo 38. Five students, Adnan, Beth, Chao, Dan, and Edmund are to be arranged in a line. How many such arrangements are possible if Beth is not allowed to stand next to Dan? (A) (B) (©) (D) €) 24 48 72 96 12039. A polygon has 12 edges. How many different diagonals does it have? (A diagonal is a line drawn from one vertex to any other vertex inside the given shape. This line cannot touch or cross any of the edges of the shape. For example, a triangle has zero diagonals and a rectangle has two.) (A) 54 (B) 66 (C) 108 (D) 132 (E) 14440. An inventory of coins contains 100 different coins. Quantity A Quantity B The number of possible collections The number of possible collections of 56 coins that can be selected of 44 coins that can be selected (the order of the coins does not (the order of the coins does not matter) matter) 41. An office supply store carries an inventory of 1,345 different products, all of which it categorizes as “business use,” “personal use,” or both. There are 740 products categorized as “business use” only and 520 products categorized as both “business use” and “personal use.” Quantity A The number of products Quantity B : 600 characterized as “personal use”42. Eight women and two men are available to serve on a committee. If three people are picked, what is the probability that the committee includes at least one man? 1 A) — (A) Ey} 1 ® | 2 O fz 5 7 (@) — 15 8 ©) = 15 43. At Lexington High School, each student studies at least one language— Spanish, French, or Latin—and no student studies all three languages. If 100 students study Spanish, 80 study French, 40 study Latin, and 22 study exactly two languages, how many students are there at Lexington High School? (A) 198 (B) 220 (Cc) 242 (D) 264 (E) 286 Of 60 birds found in a certain location, 20 are songbirds and 23 are migratory. (It is possible for a songbird to be either migratory or not migratory.) Quantity AThe number of the 60 birds that Quantity B are neither migratory nor 16 songbirds