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Show written solutions for all of the following problems. Unit 5: Counting Theory Text: A Spiral Workbook for Discrete Mathematics. 8.2: Addition and Multiplication Principles 1) Forty-six students surveyed in a film class told the professor that they had watched at least one of the three films in The Godfather trilogy. Further inquiry led to the following data: 0 41 had watched Part I; . 37 had watched Part II; . 33 had watched Part III; . 33 had watched both Parts | and II; . 30 had watched both Parts | and III; 0 29 had watched both Parts II and III. a) How many students had watched all three films? b) How many students had watched only Part I? c) How many students had watched only Part H? d) How many students had watched only Part III? 2) A professor has seven books on discrete mathematics, five on number theory, and four on abstract algebra. In how many ways can a student borrow two books not on the same subject? Hint: Which two subjects would the student choose? 3) How many different five-digit integers can be formed using the digits 1, 3, 3, 3, 5? Hint: The 35 are identical, so we cannot tell the difference between them. Consequently, what really matters is where we put the 1 and 5. Once we place the 1 and 5, the remaining three positions must be occupied by the 3s. 4) How many five-letter words (technically, we should call them strings, because we do not care if they make sense) can be formed using the letters A, B, C, and D, with repetitions allowed? How many ofthem do not contain the substring BAD? Hint: For the second question, consider using a complement. 5) Four cards are chosen at random from a standard deck of 52 playing cards, with replacement allowed. This means after choosing each card, the card is return to the deck, and the deck is reshuffled before another card is selected at random. Determine the number of such four-card sequences if a) There are no restrictions. b) None of the cards can be spades. c) All four cards are from the same suit. d) The first card is an ace and the second card is not a king. e) At least one of the four cards is an ace. 8.3: Permutations 6) a) How many functions are there from 26 to Zn? b) How many of them are one-to-one? 7) The school board of a school district has 14 members. In how many ways can the chair, first vice- chair, second vice-chair, treasurer, and secretary be selected? 8) Eleven students go to lunch. There are two circular tables in the dining hall, one can seat 7 people, the other can hold 4. In how many ways can they be seated? 8.4: Combinations 9) Becky likes to watch DVDs each evening. How many DVDs must she have if she is able to watch every evening for 24 consecutive evenings during her winter break given: a) a different subset of DVDs each evening? Hint: The power set of n elements should be more than 24. b) a different subset of three DVDs each evening? Hint: Use combinations for the subsets. 10) Bridget has 11 friends from her bridge club. Every Thursday evening, she invites three friends to her home for a bridge game. She always sits in the north position, and she decides which friends are to sit in the east, south, and west positions. She is able to do this for 200 weeks without repeating a seating arrangement. What is the minimum value of n? 11) Determine the number of permutations of {A, B, C, D, E} that satisfy the following conditions: a) A occupies the first position. b) A occupies the first position, and B the second. c) A appears before B. 12) A local pizza restaurant offers the following toppings on their cheese pizzas: extra cheese, pepperoni, mushrooms, green peppers, onions, sausage, ham, and anchovies. a) How many kinds of pizzas can one order? b) How many kinds of pizzas can one order with exactly three toppings? c) How many kinds of vegetarian pizza (without pepperoni, sausage, or harn) can one order? Text: Discrete Mathematics: An Open Introduction, Elrd Edition. 1.2 Binomial Coefficients 13) Let S = {1, 2, 3, 4, 5, 6} a) How many subsets are there of cardinality 4? Note: you can enter \"(701, k)" for G: . b) How many subsets of cardinality 4 have {2,3,5} as a subset? c) How many subsets of cardinality 4 contain at least one odd number? d) How many subsets of cardinality 4 contain exactly one even number? 14) Let A = {1, 2, 3, 9}. a) How many subsets of A are there? That is, find |50(A)|. Explain. b) How many subsets ofA contain exactly 5 elements? Explain. c) How many subsets ofA contain only even numbers? Explain. d) How many subsets ofA contain an even number of elements? Explain. 15) Suppose you are ordering a large pizza from D.P. Dough. You want 3 distinct toppings, chosen from their list of 11 vegetarian toppings. a) How many choices do you have for your pizza? b) How many choices do you have for your pizza if you refuse to have pineapple as one of your toppings? c) How many choices do you have for your pizza if you insist on having pineapple as one of your toppings? d) How do the three questions above relate to each other? Explain. 1.3 Combinations and Permutations 16) A combination lock consists of a dial with 40 numbers on it. To open the lock, you turn the dial to the right until you reach a first number, then to the left until you get to second number, then to the right again to the third number. The numbers must be distinct. How many different combinations are possible? 17) How many anagrams are there of \"anagram"? 1.5 Stars and Bars 18) Each of the counting problems below can be solved with stars and bars. Answer each question and for each, say what outcome the following diagram: \"*H | =k H H I" represents, if there are the correct number of stars and bars for the problem. Otherwise, say why the diagram does not represent any outcome, and what a correct diagram would look like. a) How many ways are there to select a handful of 6jellybeans from a jar that contains 5 different flavors? bl How many ways can you distribute 5 identical lollipops to 6 kids? c) How many 6letter words can you make using the 5 vowels in alphabetical order? d} How many solutions are there to the equation x1 + x2 + x3 + x4 = 6. 19) When playing Yahtzee, you roll five regular 6-sided dice. How many different outcomes are possible from a single roll? The order of the dice does not matter. 1.7 Chapter Summary 20) In a recent small survey of airline passengers, 25 said they had flown American in the last year, 30 had flown Jet Blue, and 20 had flown Continental. Of those, 10 reported they had flown on American and Jet Blue, 12 had flown on Jet Blue and Continental, and 7 had flown on American and Continental. 5 passengers had flown on all three airlines. How many passengers were surveyed? (Assume the results above make up the entire survey.) 21) Recall, by 8-bit strings, we mean strings of binary digits, of length 8. a) How many 8-bit strings are there total? b} How many 8-bit strings have weight 5? c] How many subsets of the set {a, b, C, d, e,f,g, h} contain exactly 5 elements? d} Explain why your answers to parts b) and c) are the same. Why are these questions equivalent? Unit 6: Probability Text: Listsl Decisions, and Graphs with an Introduction to Probabilig, Unit CL: Counting and Listing, Section 4: Probability and Basic Counting. 1) An urn contains ten balls, labeled 1, 2, . . ., 10. a) Two balls are drawn together as a pair. What is the sample space? What is the probability that the sum of the labels on the balls is odd? b} Two balls are drawn one after the other without replacement. What is the sample space? What is the probability that the sum of the labels on the balls is odd? c) Two balls are drawn one after the other with replacement. What is the sample space? What is the probability that the sum is odd? 2) You have been dealt 4 cards and discover that you have 3 of a kind; that is, 3 cards have the same face value and the fourth is different. For example, you may have been dealt 4 Q 4. IOQ 4 Q . The other three players each receive four cards, but you do not know what they have been dealt. What is the probability that the fifth card will improve your hand by making it 4 ofa kind or a full house (3 of a kind and a pair)? 3) A small deck offive cards is numbered 1 to 5. First one card and then a second card are selected at random, with replacement. What is the probability that the sum of the values on the cards is a prime number? 4) Three boys and three girls sit in a row. Find the probability that exactly two of the girls are sitting next to each other (the remaining girl separated from them by at least one boy). Text: lntroductom Probability. Chapter 4: Conditional Probability, 4.1: Discrete Conditional Probability. 5) A coin is tossed three times. What is the probability that exactly two heads occur, given that a) the first outcome was a head? b) the first outcome was a tail? cl the first two outcomes were heads? d) the first two outcomes were tails? e) the first outcome was a head and the third outcome was a head? 6) In a poker hand, John has a very strong hand and bets 5 dollars. The probability that Mary has a better hand is 0.04. If Mary had a better hand she would raise with probability 0.9, but with a poorer hand she would only raise with probability 0.1. If Mary raises, what is the probabilitythat she has a better hand than John does? Hint: Use Bayes' Theorem. You can find the formula (Theorem 1) from the "Bayes' Theorem Practice Problem" document found under the Unit 6: Probability module. Exercise 7) Suppose that 7% of the patients who tested for COVID in the US were infected. Also, suppose that when a PCR test for COVID is given, 84% of the patients infected with COVID test positive and 4% of the patients not infected with COVID test positive. Find the following probabilities where: C = the patient is infected with COVID, and T = the patient has tested positive for COVI D, given: a) a patient who tested positive for COVID with this test is infected with it. b) a patient who tested positive for COVID with this test is not infected with it. c) a patient who tested negative for COVID with this test is infected with it. d) a patient who tested negative for COVID with this test is not infected with it