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Discrete Mathematics and Its Applications 7th edition Kenneth H. Rosen - Solutions

How many ways can n books be placed on k distinguishable shelves a) If the books are indistinguishable copies of the same title? b) If no two books are the same, and the positions of the books on the shelves matter?
Use the product rule to prove Theorem 4, by first placing objects in the first box, then placing objects in the second box, and so on.
How many ways are there to assign three jobs to five employees if each employee can be given more than one job?
How many ways are there to distribute six distinguishable objects into four indistinguishable boxes so that each of the boxes contains at least one object?
How many ways are there to distribute six indistinguishable objects into four indistinguishable boxes so that each of the boxes contains at least one object?
How many ways are there to pack nine identical DVDs into three indistinguishable boxes so that each box contains at least two DVDs?
How many ways are there to distribute five balls into three boxes if each box must have at least one ball in it if a) Both the balls and boxes are labeled? b) The balls are labeled, but the boxes are unlabeled? c) The balls are unlabeled, but the boxes are labeled? d) Both the balls and boxes are
Suppose that a weapons inspector must inspect each of five different sites twice, visiting one site per day. The inspector is free to select the order in which to visit these sites, but cannot visit site X, the most suspicious site, on two consecutive days. In how many different orders can the
Prove the Multinomial Theorem: If n is a positive integer, thenWhere is a multinomial coefficient.
How many ways are there to select three unordered elements from a set with five elements when repetition is allowed?
Show that Algorithm 3 produces the next larger r-combination in lexicographic order after a given r-combination.
List all 3-permutations of {1, 2, 3, 4, 5}. The remaining exercises in this section develop another algorithm for generating the permutations of {1, 2, 3, . . . , n}. This algorithm is based on Cantor expansions of integers. Every nonnegative integer less than n! has a unique Cantor expansion a11!
Show that the correspondence described in the preamble is a bijection between the set of permutations of {1, 2, 3, . . . , n} and the nonnegative integers less than n!.
Develop an algorithm for producing all permutations of a set of n elements based on the correspondence described in the preamble to Exercise 14.
The name of a file in a computer directory consists of three uppercase letters followed by a digit, where each letter is either A, B, or C, and each digit is either 1 or 2. List the name of these files in lexicographic order, where we order letters using the usual alphabetic order of letters.
Find the next larger permutation in lexicographic order after each of these permutations. a) 1432 b) 54123 c) 12453 d) 45231 e) 6714235 f) 31528764
Use Algorithm 1 to generate the 24 permutations of the first four positive integers in lexicographic order.
Use Algorithm 3 to list all the 3-combinations of {1, 2, 3, 4, 5}.
What is meant by a combinatorial proof of an identity? How is such a proof different from an algebraic one?
a) What is the difference between an r-combination and an r-permutation of a set with n elements? b) Derive an equation that relates the number of r-combinations and the number of r-permutations of a set with n elements. c) How many ways are there to select six students from a class of 25 to serve
How many ways are there to choose 6 items from 10 distinct items when a) The items in the choices are ordered and repetition is not allowed? b) The items in the choices are ordered and repetition is allowed? c) The items in the choices are unordered and repetition is not allowed? d) The items in
A fortune cookie company makes 213 different fortunes. A student eats at a restaurant that uses fortunes from this company and gives each customer one fortune cookie at the end of a meal. What is the largest possible number of times that the student can eat at the restaurant without getting the
Show that given any set of 10 positive integers not exceeding 50 there exist at least two different five-element subsets of this set that have the same sum.
a) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces are chosen? b) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces and at least two of the 13 kinds are chosen? c) How many
Show that in a sequence of m integers there exists one or more consecutive terms with a sum divisible by m.
Show that the decimal expansion of a rational number must repeat itself from some point onward.
How many ways are there to choose a dozen donuts from 20 varieties a) If there are no two donuts of the same variety? b) If all donuts are of the same variety? c) If there are no restrictions? d) If there are at least two varieties among the dozen donuts chosen? e) If there must be at least six
Suppose that S is a set with n elements. How many ordered pairs (A, B) are there such that A and B are subsets of S with A ⊆ B?
Let n and r be integers with 1 ≤ r < n. Show that C(n, r − 1) = C(n + 2, r + 1) −2C(n + 1, r + 1) + C(n, r + 1).
How many bit strings of length n, where n ≥ 4, contain exactly two occurrences of 01?
A professor writes 20 multiple-choice questions, each with the possible answer a, b, c, or d, for a discrete mathematics test. If the number of questions with a, b, c, and d as their answer is 8, 3, 4, and 5, respectively, how many different answer keys are possible, if the questions can be placed
How many ways are there to assign 24 students to five faculty advisors?
How many subsets of a set with ten elements a) Have fewer than five elements? b) Have more than seven elements? c) Have an odd number of elements?
How many ways are there to put n identical objects into m distinct containers so that no container is empty?
Find these signless Stirling numbers of the first kind. a) c(3,2) b) c(4,2) c) c(4,3) d) c(5,4)
Show that if n is a positive integer with n ≥ 3, then c(n, n − 2) = (3n − 1)C(n, 3)/4.
How many bit strings of length 10 over the alphabet {a, b, c} have either exactly three as or exactly four bs?
Give a combinatorial proof that 2n divides n! whenever n is an even positive integer.
Suppose that when an enzyme that breaks RNA chains after each G link is applied to a 12-link chain, the fragments obtained are G, CCG, AAAG, and UCCG, and when an enzyme that breaks RNA chains after each C or U link is applied, the fragments obtained are C, C, C, C, GGU, and GAAAG. Can you
Devise an algorithm for generating all the r-permutations of a finite set when repetition is allowed.
Show that if m and n are integers with m ≥ 3 and n ≥ 3, then R(m, n) ≤ R(m, n − 1) + R(m − 1, n).
An ice cream parlor has 28 different flavors, 8 different kinds of sauce, and 12 toppings. a) In how many different ways can a dish of three scoops of ice cream be made where each flavor can be used more than once and the order of the scoops does not matter? b) How many different kinds of small
When the numbers from 1 to 1000 are written out in decimal notation, how many of each of these digits are used? a) 0 b) 1 c) 2 d) 9
Many combinatorial identities are described in this book. Find some sources of such identities and describe important combinatorial identities besides those already introduced in this book. Give some representative proofs, including combinatorial ones, of some of these identities.
Describe the latest discoveries of values and bounds for Ramsey numbers.
Describe additional ways to generate all the permutations of a set with n elements besides those found in Section 6.6. Compare these algorithms and the algorithms described in the text and exercises of Section 6.6 in terms of their computational complexity.
What is the probability that a five-card poker hand contains the two of diamonds, the three of spades, the six of hearts, the ten of clubs, and the king of hearts?
What is the probability that a five-card poker hand contains at least one ace?
What is the probability that a five-card poker hand contains two pairs (that is, two of each of two different kinds and a fifth card of a third kind)?
What is the probability that a five-card poker hand contains a straight, that is, five cards that have consecutive kinds?
What is the probability that a five-card poker hand contains cards of five different kinds and does not contain a flush or a straight?
What is the probability that a fair die never comes up an even number when it is rolled six times?
What is the probability that a positive integer not exceeding 100 selected at random is divisible by 5 or 7?
Find the probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding a) 50. b) 52. c) 56. d) 60.
Find the probability of selecting exactly one of the correct six integers in a lottery, where the order in which these integers are selected does not matter, from the positive integers not exceeding a) 40. b) 48. c) 56. d) 64.
In a super lottery, players win a fortune if they choose the eight numbers selected by a computer from the positive integers not exceeding 100. What is the probability that a player wins this super lottery?
Suppose that 100 people enter a contest and that different winners are selected at random for first, second, and third prizes. What is the probability that Michelle wins one of these prizes if she is one of the contestants?
What is the probability that Abby, Barry, and Sylvia win the first, second, and third prizes, respectively, in a drawing if 200 people enter a contest and a) No one can win more than one prize. b) Winning more than one prize is allowed.
In roulette, a wheel with 38 numbers is spun. Of these, 18 are red, and 18 are black. The other two numbers, which are neither black nor red, are 0 and 00. The probability that when the wheel is spun it lands on any particular number is 1/38 a) What is the probability that the wheel lands on a red
Which is more likely: rolling a total of 9 when two dice are rolled or rolling a total of 9 when three dice are rolled?
Explain what is wrong with the statement that in the Monty Hall Three-Door Puzzle the probability that the prize is behind the first door you select and the probability that the prize is behind the other of the two doors that Monty does not open are both 1/2, because there are two doors left.
This problem was posed by the Chevalier de Méré and was solved by Blaise Pascal and Pierre de Fermat. a) Find the probability of rolling at least one six when a fair die is rolled four times. b) Find the probability that a double six comes up at least once when a pair of dice is
What is the probability that the sum of the numbers on two dice is even when they are rolled?
What is the probability that when a coin is flipped six times in a row, it lands heads up every time?
What is the probability that a five-card poker hand does not contain the queen of hearts?
What probability should be assigned to the outcome of heads when a biased coin is tossed, if heads is three times as likely to come up as tails? What probability should be assigned to the outcome of tails?
Show that if E and F are events, then p (E ∩ F) ≥ p(E) + p(F) − 1. This is known as Bonferroni's inequality.
Show that if E1, E2, . . . , En are events from a finite sample space, then p(E1 ∪ E2 ∪ · · · ∪ En) ≤ p(E1) + p(E2)+· · ·+p(En). This is known as Boole's inequality
If E and F are independent events, prove or disprove that E and F are necessarily independent events. In Exercises 18, 20, and 21 assume that the year has 366 days and all birthdays are equally likely. In Exercise 19 assume it is equally likely that a person is born in any given month of the year.
a) What is the probability that two people chosen at random were born during the same month of the year? b) What is the probability that in a group of n people chosen at random, there are at least two born in the same month of the year? c) How many people chosen at random are needed to make the
Find the smallest number of people you need to choose at random so that the probability that at least two of them were both born on April 1 exceeds 1/2.
What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up heads?
What is the conditional probability that a randomly generated bit string of length four contains at least two consecutive 0s, given that the first bit is a 1? (Assume the probabilities of a 0 and a 1 are the same.)
Let E and F be the events that a family of n children has children of both sexes and has at most one boy, respectively. Are E and F independent if a) n = 2? b) n = 4? c) n = 5?
A group of six people play the game of "odd person out" to determine who will buy refreshments. Each person flips a fair coin. If there is a person whose outcome is not the same as that of any other member of the group, this person has to buy the refreshments. What is the probability that there is
Find the probability of each outcome when a biased die is rolled, if rolling a 2 or rolling a 4 is three times as likely as rolling each of the other four numbers on the die and it is equally likely to roll a 2 or a 4.
Find the probability that a family with five children does not have a boy, if the sexes of children are independent and if a) A boy and a girl are equally likely. b) The probability of a boy is 0.51. c) The probability that the ith child is a boy is 0.51 − (i/100).
Find the probability that the first child of a family with five children is a boy or that the last two children of the family are girls, for the same conditions as in parts (a), (b), and (c) of Exercise 31.In Exercise 31a) A boy and a girl are equally likely.b) The probability of a boy is 0.51.c)
Find each of the following probabilities when n independent Bernoulli trials are carried out with probability of success p. a) The probability of no failures b) The probability of at least one failure c) The probability of at most one failure d) The probability of at least two failures
Use pseudocode to write out the probabilistic primality test described in Example 16.
A pair of dice is loaded. The probability that a 4 appears on the first die is 2/7, and the probability that a 3 appears on the second die is 2/7. Other outcomes for each die appear with probability 1/7. What is the probability of 7 appearing as the sum of the numbers when the two dice are rolled?
What is the probability of these events when we randomly select a permutation of {1, 2, 3, 4}? a) 1 precedes 4. b) 4 precedes 1. c) 4 precedes 1 and 4 precedes 2. d) 4 precedes 1, 4 precedes 2, and 4 precedes 3. e) 4 precedes 3 and 2 precedes 1.
What is the probability of these events when we randomly select a permutation of the 26 lowercase letters of the English alphabet? a) The permutation consists of the letters in reverse alphabetic order. b) z is the first letter of the permutation. c) z precedes a in the permutation. d) a
Suppose that E and F are events in a sample space and p(E) = 1/3, p(F) = 1/2, and p(E | F) = 2/5. Find p(F | E).
An electronics company is planning to introduce a new camera phone. The company commissions a marketing report for each new product that predicts either the success or the failure of the product. Of new products introduced by the company, 60% have been successes. Furthermore, 70% of their
Suppose that E, F1, F2, and F3 are events from a sample space S and that F1, F2, and F3 are pair wise disjoint and their union is S. Find p(F1 | E) if p(E | F1) = 1/8, p(E | F2)=1/4, p(E | F3)=1/6, p(F1)=1/4, p(F2)= 1/4, and p(F3) = 1/2.
In this exercise we will use Bayes' theorem to solve the Monty Hall puzzle (Example 10 in Section 7.1). Recall that in this puzzle you are asked to select one of three doors to open. There is a large prize behind one of the three doors and the other two doors are losers. After you select a door,
Suppose that a Bayesian spam filter is trained on a set of 1000 spam messages and 400 messages that are not spam. The word "opportunity" appears in 175 spam messages and 20 messages that are not spam. Would an incoming message be rejected as spam if it contains the word "opportunity" and the
Suppose that a Bayesian spam filter is trained on a set of 10,000 spam messages and 5000 messages that are not spam. The word "enhancement" appears in 1500 spam messages and 20 messages that are not spam, while the word "herbal" appears in 800 spam messages and 200 messages that are not spam.
Suppose that E1 and E2 are the events that an incoming mail message contains the words w1 and w2, respectively. Assuming that E1 and E2 are independent events and that E1 | S and E2 | S are independent events, where S is the event that an incoming message is spam, and that we have no prior
Suppose that Frida selects a ball by first picking one of two boxes at random and then selecting a ball from this box at random. The first box contains two white balls and three blue balls, and the second box contains four white balls and one blue ball. What is the probability that Frida picked a
Suppose that 8% of all bicycle racers use steroids, that a bicyclist who uses steroids tests positive for steroids 96% of the time, and that a bicyclist who does not use steroids tests positive for steroids 9% of the time. What is the probability that a randomly selected bicyclist who tests
Suppose that a test for opium use has a 2% false positive rate and a 5% false negative rate. That is, 2% of people who do not use opium test positive for opium, and 5% of opium users test negative for opium. Furthermore, suppose that 1% of people actually use opium. a) Find the probability that
Suppose that 8% of the patients tested in a clinic are infected with HIV. Furthermore, suppose that when a blood test for HIV is given, 98% of the patients infected with HIV test positive and that 3% of the patients not infected with HIV test positive. What is the probability that a) A patient
Suppose that we roll a fair die until a 6 comes up or we have rolled it 10 times. What is the expected number of times we roll the die?
Suppose that we roll a pair of fair dice until the sum of the numbers on the dice is seven. What is the expected number of times we roll the dice?
Show that if the random variable X has the geometric distribution with parameter p, and j is a positive integer, then p(X ≥ j) = (1 − p) j−1.
Estimate the expected number of integers with 1000 digits that need to be selected at random to find a prime, if the probability a number with 1000 digits is prime is approximately 1/2302.
Let X be the number appearing on the first die when two fair dice are rolled and let Y be the sum of the numbers appearing on the two dice. Show that E(X) E(Y) ≠ E(XY).
What is expected value of the sum of the numbers appearing on two fair dice when they are rolled given that the sum of these numbers is at least nine? That is, what is E(X | A) where X is the sum of the numbers appearing on the two dice and A is the event that X ‰¥ 9? The law of total

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