# 6. In how many different ways can you answer 10 multiple-choice questions if each question has five choices?

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1 Pre-Calculus Section 4.1 Multiplication, Addition, and Complement 1. Evaluate each of the following: a. 5! b. 6! c. 7! d. 0! 2. Evaluate each of the following: a. 10! b. 20! 9! 18! 3. In how many different ways can your arrange five books on a shelf? 4. In how many different ways can nine people be arranged in a line? 5. In how many different ways can you answer 10 true-false questions? 6. In how many different ways can you answer 10 multiple-choice questions if each question has five choices? 7. Many radio stations have 4-letter call signs beginning with K. How many such call signs are possible if: a. letters can be repeated? b. letters cannot be repeated? 8. How many 3-digit numbers can be formed using the digits 4, 5, 6, 7, 8 if: a. the digits can be repeated? b. the digits cannot be repeated? 9. In how many ways can four people be seated in a row of 12 chairs? 10. In how many ways can four different prizes be given to any four of ten people if no person receives more than one prize? 11. Four cards numbered 1 through 4 are shuffled and three different cards are chosen one at a time. How many ways can this be done? 12. A high school coach must decide on the batting order for a baseball team of nine players. a. How many different batting orders are possible? b. How many different batting orders are possible if the pitcher bats last? c. How many different batting orders are possible if the pitcher bats last and the team s best hitter bats third? 13. A track coach must choose a 4-person 400 m relay team and a 4-person 800 m relay team from a squad of seven sprinters, any of whom can run on either team. If the fastest sprinter runs last in both races, in how many ways can the coach form the two teams if each of the six remaining sprinters runs only once and each different order is counted as a different team? 14. How many numbers consisting of one, two or three digits (without repetitions) can be formed using the digits 1, 2, 3, 4, 5, 6? 15. If you have five signal flags and can send messages by hoisting one or more flags on a flagpole, how many messages can you send? 16. In some states license plates consist of three letters followed by two or three digitis (for example, RRK54 or ABC055). How many such possibilities are there for: a. those plates with two digits? b. those plates with three digits? c. In all, how many license plates are possible? 17. How many possibilities are there for a license plate with two letters and three or four non-zero digits? 18. a. How many three-digit numbers contain no 7 s? b. How many three-digit numbers contain at least one 7?

2 19. a. How many four-digit numbers contain no 8 s or 9 s? b. How many four-digit numbers contain at least one 8 or 9? 20. How many numbers from 5000 to 6999 contain at least one 3? 21. Many license plates in the U.S. consist of three letters followed by a three-digit number from 100 to 999. How many of these contain at least one of the vowels A, E, I, O, and U? 22. Telephone numbers in the U.S. and Canada have 10 digits as follows: Three-digit area code number: first digit is not 0 or 1 Second digit must be 0 or 1 Three-digit exchange number: first and second digits are not 0 or 1 Four-digit line number not all zeros a. How many possible area codes are there? b. The area code for Frankfort is 815. Within this area code how many exchange numbers are possible? c. The area code for Tinley Park is 708. Within this exchange, how many line numbers are possible? d. How many seven-digit phone numbers are possible in the 815 area code? e. How many ten-digit phone numbers are possible in the U.S. and Canada? f. Suppose a state has five telephone area codes. How many phone numbers could there be in the state without adding any more area codes? 23. a. How many nine-letter words can be formed using the letters of the word FISHERMAN? b. How many nine-letter words begin and end with a vowel? 24. Suppose the letters of VERMONT are used to form words. a. How many seven-letter words can be formed? b. How many six-letter words can be formed? c. how many five letter words can be formed which begin with a vowel and end with a consonant? Pre-Calculus Section 4.2 Permutations and Combinations 1. a. In how many ways can a club with 20 members choose a president and a vice-president? b. In how many ways can the club choose a two-person governing council? 2. a. In how many ways can a club with 13 members choose four different officers? b. In how many ways can the club choose a four-person governing council? 3. a. In how many ways can a host-couple choose four couples to invite for dinner from a group of ten couples? b. Ten students each submit a woodworking project in an industrial arts competition. There is to be a first-place prize, a second-place prize, a third-place prize, and an honorable mention. In how many ways can these awards be presented? 4. A teach has a collection of 20 true-false questions and wishes to choose five of them for a quiz. How many quizzes can be made if: a. the order of the questions on the quiz is important? b. the order of the questions on the quiz is unimportant? 5. Each of the 200 students attending a school dance has a ticket with a number for a door prize. If three different numbers are selected, how many ways are there to award the prizes: a. given that the prizes are identical? b. given that the prizes are different? 6. Suppose you bought four books and gave one to each of four friends. In how many ways can the books be given if: a. they are all different? b. they are all identical? 7. Eight people apply for three job positions. In how many different ways can the three positions be filled if: a. the positions are all different? b. the positions are all the same?

3 8. How many different ways are there to deal a hand of five cards from a standard deck of 52 cards if: a. the order in which the cards are dealt is important? b. the order in which the cards are dealt is not important? 9. In how many ways can six hockey players be chosen from a group of twelve if: a. the playing positions are considered? b. the playing positions are not considered? 10. Of the twelve players on a school s basketball team, the coach must choose five players to be in the starting lineup. In how many ways can this be done if: a. the playing positions are considered? b. the playing positions are not considered? 11. a. A hiker would like to invite seven friends to go on a trip but has room for only four of them. In how many ways can they be chosen? b. If there were room for only three friends, in how many ways could they be chosen? 12. a. In how many ways can you choose three letters from the word LOGARITHM if the order of the letters is unimportant? b. In how many ways can you choose six letters from the word LOGARITHM if the order of the letters is unimportant? 13. A certain chain of ice cream stores sells 28 different flavors, and a customer can order a single-scoop, double-scoop, or triple-scoop cone. Suppose on a multi-scoop cone that the order of the flavors is important and that the flavors can be repeated. How many possible cones are there? 14. Three couples go to the movies and sit together in a row of six seats. In how many ways can these people arrange themselves if each couple sits together? 15. A math teacher uses four algebra books, two geometry books, and three pre-calculus books for reference. In how many ways can the teacher arrange the books on a shelf if books covering the same subject matter are kept together? Pre-Calculus Section 4.3 Introduction to Probability 1. Suppose a card is drawn from a well-shuffled standard deck of 52 cards. Find the probability of drawing each of the following: a. a black card b. a spade c. not a spade d. a black face card e. a black jack f. not a black jack g. a red diamond h. a black diamond i. not a black diamond j. a jack k. a jack or king l. neither a jack nor a king 2. Mr. and Mrs. Smith each bought ten raffle tickets. Each of their three children bought four tickets. If 4280 tickets were sold in all, what is the probability that the grand prize winner is: a. Mr. or Mrs. Smith? b. one of the 5 Smiths? c. none of the Smiths? 3. One of the integers between 11 and 20 inclusive, is picked at random. What is the probability that: a. the integer is even? b. the integer is divisible by 3? c. the integer is prime? 4. Two fair dice are rolled. Find the probability that: a. the sum is 6. b. the sum is 7. c. the sum is 8. d. the sum is even e. the sum is 12. f. the sum is less than 12. g. the two dice show different numbers h. the red die shows a greater number than the white die. 5. A die is rolled and a coin is tossed. Find the probability that the die s number is even and the coin is heads. 6. The numbers 1, 2, 3, and 4 are written on separate slips of paper and placed in a hat. Two slips of paper are then randomly drawn, one after the other and without replacement. Find the probability that the sum of the numbers picked is 6 or more.

4 7. Suppose that the odds in favor of the National League s winning the All-Star Gram are 4:3. a. What is the probability that the National League wins? b. What is the probability that the American League wins? 8. Suppose that the odds in favor of the incumbent s winning an election are 2:3. a. What is the probability that the incumbent wins? b. What is the probability that the incumbent s challenger wins? 9. Suppose you roll two dice, each of which is a regular octahedron with faces numbered 1 to 8. a. What is the probability that the sum of the numbers showing is 2? b. What is the probability that the sum of the numbers showing is 3? c. What sum is most likely to appear? 10. Suppose you roll two dice, each of which is a regular dodecahedron with faces numbered 1 to 12. a. What is the probability that the sum of the numbers showing is 24? b. What is the probability that the sum is 23? c. What sum is most likely to appear? 11. From a group consisting of Alvin, Bob, Carol, and Donna, two people are to be randomly selected to serve on a committee. a. Find the probability that Bob and Carol are selected. b. Find the probability that Carol is not selected. 12. From a standard deck of cards, two cards are randomly drawn, one after the other without replacement. The color of each card is noted. Which is less likely to occur: two red cards or a red card and a black card? Explain. Pre-Calculus Section 4.4 Probability with Combinations 1. Five cards are dealt from a well-shuffled standard deck. a. Using combinations, write an expression for the probability all are hearts. b. Using combinations, write an expression for the probability there are no hearts. c. Using combinations, write an expression for the probability there is at least one heart. 2. Thirteen cards are dealt from a well-shuffled standard deck. a. Using combinations, write an expression for the probability all are clubs. b. Using combinations, write an expression for the probability none are clubs. c. Using combinations, write an expression for the probability there is at least one club. 3. A bag contains five red marbles and three white marbles. If two marbles are randomly drawn, one after the other and without replacement, what is the probability that there is one red marble? 4. A bag contains six red marbles and two white marbles. If two marbles are randomly drawn, one after the other and without replacement, what is the probability that there is one red marble? 5. Free concert tickets are distributed to four students chosen at random from eight juniors and twelve seniors in the school orchestra. What is the probability that free tickets are received by: a. four seniors? b. exactly three seniors? c. exactly two seniors? d. exactly one senior? e. no seniors? 6. A town council consists of eight Democrats, seven Republicans, and five Independents. A committee of three is chosen by randomly pulling names from a hat. a. What is the probability that the committee has two Democrats and one Republican? b. What is the probability that the committee has three Independents? c. What is the probability that the committee has no Independents? d. What is the probability that the committee has one Democrat, one Republican, and one Independent?

5 7. A committee of three people is to be randomly selected from the six people: Archibald, Beatrix, Charlene, Denise, Eloise, and Fernando. a. Find the probability that Eloise is on the committee. b. Find the probability that Eloise and Fernando are on the committee. c. Find the probability that either Eloise or Fernando is on the committee. d. Find the probability that neither Eloise nor Fernando is on the committee. e. Find the probability that Archibald is on the committee and Beatrix is not. f. Find the probability that Denise is on the committee given that Archibald is. g. Find the probability that Denise is on the committee given that neither Archibald nor Beatrix is. 8. Thirteen cards are dealt from a well-shuffled standard deck. a. Using combinations, write an expression for the probability of getting all red cards. b. Using combinations, write an expression for the probability of getting seven diamonds and six hearts. c. Using combinations, write an expression for the probability of getting at least one face card. d. Using combinations, write an expression for the probability of getting all face cards. e. Using combinations, write an expression for the probability of getting all cards form the same suit. f. Using combinations, write an expression for the probability of getting seven spades, three hearts, and three clubs. g. Using combinations, write an expression for the probability of getting all of twelve face cards. h. Using combinations, write an expression for the probability of getting at least one diamond. 9. A committee of four is chosen at random from a group of five married couples. What is the probability that the committee includes no two people who are married to each other? 10. The letters of the word ABRACADABRA are written on separate slips of paper and place din a hat. Five slips of paper are then randomly drawn, one after the other and without replacement. a. What is the probability of getting all A s? b. What is the probability of getting both R s? c. What is the probability of getting at least one B? d. What is the probability of getting at least one of each letter? 11. A quality control inspector randomly inspects four microchips in every lot of 100. If one or more microchips are defective, the entire lot is rejected for shipment. Suppose a lot contains 10 defective microchips and 90 acceptable ones. What is the probability that the lot is rejected? Round your answer to three decimal places. 12. A lot of 20 television sets consists of six defective sets and fourteen good ones. If a sample of three sets is chosen, a. What is the probability the sample contains all defective sets? b. What is the probability that the sample contains at least one defective set? Pre-Calculus Unit 4 Review 1. On a bookshelf there are 10 different algebra books, six different geometry books, and four different calculus books. In how many ways can you choose three books, one of each kind? 2. A standard bridge deck of cards consists of four suits of thirteen cards each. How many five card hands can be dealt that includes four cards from the same suit and one card from a different suit? 3. In how many different ways can a 10-question true-false test be answered if each question much be answered? 4. At Hamburger Heaven, you can order hamburgers with cheese, onion, pickle, relish, mustard, lettuce and/or tomato. How many different combinations of the extras can you order, choosing any three of them? 5. How many ways are there of selecting three cards, with replacement, from a deck of 26 cards? 6. Students are required to answer any eight out of ten questions on a certain test. How many different combinations of eight questions can a student choose to answer? 7. A student council has six seniors, five juniors, and one sophomore as members. In how many ways can a three member council committee be formed that includes one member of each class? 8. How many license plates of five symbols can be made using a letter for the first symbol and digits for the remaining four symbols?

6 9. How many three letter subsets can be formed from the set {P, Q, R, S}? 10. From a group of three men and seven women, how many committees of two men and two women can be formed? 11. There are five persons who are applicants for three different positions in a store, each person being qualified for each position. In how many ways is it possible to fill the positions, if each person can only fill one position? 12. A bag contains four red and six white marbles. How many ways can five marbles be selected if exactly two must be red? 13. The letters r, s, t, u, and v are to be used to form five-letter patterns. How many patterns can be formed if a. repetitions are not allowed? b. repetitions are allowed? 14. If a number having six digits is to be built at random by use of the digits 1, 2, 3, 4, 5, 6, 7 without repetition, find the number of ways the number could be: a. odd b. even c. divisible by five d. greater than 500,000 e. either odd or even 15. What is the probability of getting a total of 6 on a roll of a pair of dice? 16. From a bag containing six nickels, ten dimes, and four quarters. Six coins are drawn at random all at once. What is the probability of getting three nickels, two dimes and one quarter? 17. If a basketball player has a free throw average of 0.80, what is the probability that the player will make two freethrows in a row? 18. There are 52 colored balls in a large tumbler: 13 red, 13 blue, 13 yellow, and 13 green. The balls of each color are lettered A through M. Five balls are chosen at random. a. What is the probability of choosing two of the same letter and three other, different letters? b. What is the probability of choosing two of the same letter, two of a second letter, and the fifth of a different letter? 19. A die is rolled twice. What is the probability of rolling a six on the first roll and a two on the second roll? 20. Dave, Tom, Bob, Roger, and Sergio left their hats at the hat-check room in a restaurant. When they asked for their hats they found that their hats had not been marked. Their hats were returned to them at random. What is the probability that they all received the correct hats? 21. What is the probability of drawing a king or a heart from a well-shuffled deck of cards in one draw? 2. A bag contains four red marbles and six green marbles. A marble is drawn and then replaced. A second drawing is made. What is the probability that a green marble is drawn both times? 23. A bag contains four red marbles and six green marbles. Two marbles are drawn together at random. What is the probability that both marbles are green? 24. If a bag contains five red balls and seven black balls, find the probability of drawing a red ball on one random draw. 25. A bag contains five red and six white balls. If we draw four balls together, find the probability that: a. two are red and two are white b. all are of the same color c. at least three are white

7 26. The probability that a certain man will live ten years is ¼, and that his wife will live ten years is 1/5. Find the probability that: a. both will live 10 years b. the man will live and the wife will not live ten years c. both will die before the end of 10 years d. the man will not live and the wife will live ten years. e. At least one of them will live ten years. f. What is the sum of the answers for parts a, b, c, d? Why? 27. If two dice are thrown what is the probability of throwing a total of seven or eleven? 28. If four men and four women are seated in a row, what is the probability the men and women alternate? 29. If a family of four takes seats at random in a row, what is the probability the father and mother a. sit together? b. Sit on the ends? 30. From five men and seven women, a committee of four is to be chosen by lot. Find the probability that the committee will involve a. four men b. two men and two women c. all men or all women d. at least three women

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