Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
8th Edition
ISBN: 9781259676512
Author: Kenneth H Rosen
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 6.4, Problem 1E
Find the expansion of (r +
- using combinatorial reasoning, as inExample1.
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Q1. A group of five applicants for a pair of identical jobs consists of three men and two
women. The employer is to select two of the five applicants for the jobs. Let S
denote the set of all possible outcomes for the employer's selection. Let A denote
the subset of outcomes corresponding to the selection of two men and B the subset
corresponding to the selection of at least one woman. List the outcomes in A, B,
AUB, AN B, and An B. (Denote the different men and women by M₁, M2, M3
and W₁, W2, respectively.)
For the following function, find the full power series centered at a
of convergence.
0 and then give the first 5 nonzero terms of the power series and the open interval
=
f(2) Σ
8
1(x)--(-1)*(3)*
n=0
₤(x) = + + + ++...
The open interval of convergence is:
1
1
3
f(x)=
=
28
3x6 +1
(Give your answer in help (intervals) .)
Q3 (8 points)
Q3. A survey classified a large number of adults according to whether they were diag-
nosed as needing eyeglasses to correct their reading vision and whether they use
eyeglasses when reading. The proportions falling into the four resulting categories
are given in the following table:
Use Eyeglasses for Reading
Needs glasses Yes
No
Yes
0.44
0.14
No
0.02
0.40
If a single adult is selected from the large group, find the probabilities of the events
defined below. The adult
(a) needs glasses.
(b) needs glasses but does not use them.
(c) uses glasses whether the glasses are needed or not.
Chapter 6 Solutions
Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
Ch. 6.1 - There are 18 mathematics majors and 325 computer...Ch. 6.1 - An office building contains 27 floors and has 37...Ch. 6.1 - A multiple-choice test contains 10 questions....Ch. 6.1 - A particular of shirt comes in 12 colors, has a...Ch. 6.1 - Six different fly from New York to Denver and...Ch. 6.1 - There are four major auto routes from Boston to...Ch. 6.1 - How many different three-letter initials can...Ch. 6.1 - How many different three-letter initials with none...Ch. 6.1 - How many different three-letter initials are there...Ch. 6.1 - How many bit strings are there of length eight?
Ch. 6.1 - How many bit strings of length ten both begin and...Ch. 6.1 - How many bit strings are there of length six or...Ch. 6.1 - How many bit strings with length not exceeding n,...Ch. 6.1 - How many bit strings of lengthn,wherenis a...Ch. 6.1 - How many strings are there of lowercase letters of...Ch. 6.1 - How many strings are there of four lowercase...Ch. 6.1 - How many strings of five ASCII characters @ (“at”...Ch. 6.1 - How many 5-element DNA sequences end with A? start...Ch. 6.1 - lg.How many 6-element RNA sequences Do not contain...Ch. 6.1 - How many positive integers between 5 and 31 are...Ch. 6.1 - How many positive integers between 50 and 100 are...Ch. 6.1 - How many positive integers less than 1000 are...Ch. 6.1 - How many positive integers between 100 and 999...Ch. 6.1 - How many positive integers between 1000 and 9999...Ch. 6.1 - How many strings of three decimal digits do not...Ch. 6.1 - How many strings of four decimal digits do not...Ch. 6.1 - Prob. 27ECh. 6.1 - How many license, plates can be made using either...Ch. 6.1 - How many license plates can be made using either...Ch. 6.1 - How many license plates can be made using either...Ch. 6.1 - How many license plates can be made using either...Ch. 6.1 - How many strings of eight uppercase English...Ch. 6.1 - How many strings of eight English letters are...Ch. 6.1 - Prob. 34ECh. 6.1 - How many one-to-one functions are there from a set...Ch. 6.1 - How many functions are there from the set {1,2,n},...Ch. 6.1 - Prob. 37ECh. 6.1 - How many partial functions (seeSection 2.3)are...Ch. 6.1 - Prob. 39ECh. 6.1 - Prob. 40ECh. 6.1 - Prob. 41ECh. 6.1 - How many 4-element DNA sequences do not contain...Ch. 6.1 - How many 4-eJement RNA sequenoes contain the base...Ch. 6.1 - On each of the 22 work days in a particular month,...Ch. 6.1 - At a large university, 434 freshman, 883...Ch. 6.1 - Prob. 46ECh. 6.1 - How many ways are there to seat six people around...Ch. 6.1 - In how many ways can a photographer at a wedding...Ch. 6.1 - In how many ways can a photographer at a wedding...Ch. 6.1 - How many bit strings of length seven either begin...Ch. 6.1 - Prob. 51ECh. 6.1 - How many bit strings of length 10 contain either...Ch. 6.1 - How many bit strings of length eight contain...Ch. 6.1 - ...Ch. 6.1 - Prob. 55ECh. 6.1 - Prob. 56ECh. 6.1 - Suppose that a password for a computer system must...Ch. 6.1 - The name, of a variable in the C programming...Ch. 6.1 - The name of a variable in the JAVA programming...Ch. 6.1 - 6o, The International Telecommunications Union...Ch. 6.1 - Prob. 61ECh. 6.1 - A key in the Vigenere cryptosystem is a string of...Ch. 6.1 - Prob. 63ECh. 6.1 - Suppose that P and q are prime numbers and than n...Ch. 6.1 - Use the principle of inclusion-exclusion to find...Ch. 6.1 - Prob. 66ECh. 6.1 - Prob. 67ECh. 6.1 - Prob. 68ECh. 6.1 - Prob. 69ECh. 6.1 - Prob. 70ECh. 6.1 - Prob. 71ECh. 6.1 - Determine the number of matches played in a...Ch. 6.1 - Prob. 73ECh. 6.1 - *74-Use the product rule to show that there are 22...Ch. 6.1 - Prob. 75ECh. 6.1 - Use mathematical induction to prove the product...Ch. 6.1 - Prob. 77ECh. 6.1 - Prob. 78ECh. 6.2 - Prob. 1ECh. 6.2 - Show that if there are 30 students in a class,...Ch. 6.2 - A drawer contains a dozen brown socks and a dozen...Ch. 6.2 - Abowl contains 10 red balls and 10 blue balls....Ch. 6.2 - Undergraduate students at a college belong to one...Ch. 6.2 - 6,There are six professors teaching the...Ch. 6.2 - group of five (not necessarily consecutive)...Ch. 6.2 - 8,Let d be a positive integer, Show that among anv...Ch. 6.2 - Letnbe a positive integer. Show that in any set...Ch. 6.2 - Prob. 10ECh. 6.2 - Prob. 11ECh. 6.2 - Prob. 12ECh. 6.2 - Prob. 13ECh. 6.2 - Prob. 14ECh. 6.2 - Show that if five integers are selected from the...Ch. 6.2 - i6. Show that if seven integers are selected from...Ch. 6.2 - How many numbers must be selected from the set...Ch. 6.2 - Howmany numbers must be selected from the set...Ch. 6.2 - A company stores products in a warehouse. Storage...Ch. 6.2 - Suppose that there are nine students in a discrete...Ch. 6.2 - i. Suppose that every student in a discrete...Ch. 6.2 - Prob. 22ECh. 6.2 - Construct a sequenceof16 positive integers that...Ch. 6.2 - Prob. 24ECh. 6.2 - Show that whenever 25 girl? and 25 boys are seated...Ch. 6.2 - Prob. 26ECh. 6.2 - Descnbe an algorithm in pseudocode for producing...Ch. 6.2 - Prob. 28ECh. 6.2 - Prob. 29ECh. 6.2 - Prob. 30ECh. 6.2 - Prob. 31ECh. 6.2 - Prob. 32ECh. 6.2 - Prob. 33ECh. 6.2 - Prob. 34ECh. 6.2 - In the 17th century, there were more than 800,000...Ch. 6.2 - Prob. 36ECh. 6.2 - Prob. 37ECh. 6.2 - Prob. 38ECh. 6.2 - A computer network consists of six computers, Each...Ch. 6.2 - Prob. 40ECh. 6.2 - Prob. 41ECh. 6.2 - Prob. 42ECh. 6.2 - Ad arm wrestler is the champion for a period of 75...Ch. 6.2 - Prob. 44ECh. 6.2 - Prob. 45ECh. 6.2 - ,There are 51 houses on a street, Each house has...Ch. 6.2 - Letibe an irrational number, Showthatfor some...Ch. 6.2 - Prob. 48ECh. 6.2 - Prob. 49ECh. 6.3 - i. List all the permutations of{a, b,c}.Ch. 6.3 - How many different permutations are there of the...Ch. 6.3 - How many permutations of{a, b,c, d,e.fg]end withCh. 6.3 - LetS = {i,2, 3,4, 5}. List all the 3-permutations...Ch. 6.3 - Find the value of each of these quantities P(6,3)...Ch. 6.3 - Find the value of each of these quantities. CCs,i)...Ch. 6.3 - Find the number of 5-permutations of a set Kith...Ch. 6.3 - In how many different orders can five runners...Ch. 6.3 - Prob. 9ECh. 6.3 - There are six different candidates for governor of...Ch. 6.3 - ii.How many bit strings of length 10 contain...Ch. 6.3 - IE.How many bit strings of length12contain exactly...Ch. 6.3 - A group contains n men and n women. How many ways...Ch. 6.3 - Prob. 14ECh. 6.3 - Prob. 15ECh. 6.3 - Prob. 16ECh. 6.3 - Prob. 17ECh. 6.3 - Prob. 18ECh. 6.3 - Prob. 19ECh. 6.3 - Prob. 20ECh. 6.3 - Prob. 21ECh. 6.3 - Prob. 22ECh. 6.3 - Prob. 23ECh. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Prob. 26ECh. 6.3 - Prob. 27ECh. 6.3 - Prob. 28ECh. 6.3 - Prob. 29ECh. 6.3 - Prob. 30ECh. 6.3 - Prob. 31ECh. 6.3 - Prob. 32ECh. 6.3 - Prob. 33ECh. 6.3 - Prob. 34ECh. 6.3 - Prob. 35ECh. 6.3 - Prob. 36ECh. 6.3 - Prob. 37ECh. 6.3 - Prob. 38ECh. 6.3 - Prob. 39ECh. 6.3 - Prob. 40ECh. 6.3 - Prob. 41ECh. 6.3 - Find the number of circular 3-permutations...Ch. 6.3 - Prob. 43ECh. 6.3 - Prob. 44ECh. 6.3 - How many ways are there for a horse race with...Ch. 6.4 - Find the expansion of (r + using combinatorial...Ch. 6.4 - Find the expansion of Cr + j,)5 using...Ch. 6.4 - Find the expansionCh. 6.4 - Find the coefficient of in Cr + y)13.Ch. 6.4 - How many terms are therein the expansion of...Ch. 6.4 - What isthecoefficient of .v in (1 +1)Ch. 6.4 - What is the coefficient of i9 in (2 - 1)Ch. 6.4 - What is the coefficient ofxsy9 in the expansion of...Ch. 6.4 - What is the coefficient of xloly" in the expansion...Ch. 6.4 - Prob. 10ECh. 6.4 - Prob. 11ECh. 6.4 - IS. Use the binomial theorem to find the...Ch. 6.4 - *3-Use the binomial theorem to find the...Ch. 6.4 - Give a formula for the coefficient ofi^in the...Ch. 6.4 - Prob. 15ECh. 6.4 - The row of Pascal’s triangle containing the...Ch. 6.4 - What is the r ow of Pascal's triangle containing...Ch. 6.4 - Prob. 18ECh. 6.4 - Prob. 19ECh. 6.4 - so. Use Exercise 18 andCorollary 1to show that...Ch. 6.4 - Prob. 21ECh. 6.4 - Suppose thatbis an integer withb> 7. Use the...Ch. 6.4 - Prove Pas cal’s identity, u sing the formula for...Ch. 6.4 - Suppose that t andnare integers withi which...Ch. 6.4 - Provethatifnandfcareintegers^th i< fc using a...Ch. 6.4 - Prove the identity (")(') = (J)(Xf), whenever n,...Ch. 6.4 - Prob. 27ECh. 6.4 - Prob. 28ECh. 6.4 - Letnbe a positive integer. Show thatCh. 6.4 - Prob. 30ECh. 6.4 - Prove the hockey-stick identity ('?’)...Ch. 6.4 - Show that if ra is a positive integer, then =2t" i...Ch. 6.4 - Prob. 33ECh. 6.4 - Prob. 34ECh. 6.4 - Prob. 35ECh. 6.4 - Prove the binomial theorem using mathematical...Ch. 6.4 - In this exercise we will count the number of paths...Ch. 6.4 - Prob. 38ECh. 6.4 - Prob. 39ECh. 6.4 - Prob. 40ECh. 6.4 - Prob. 41ECh. 6.4 - Prob. 42ECh. 6.4 - Determine a formula involving binomial...Ch. 6.5 - In how many different wavs can five elements be...Ch. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - How many different ways are there to choose a...Ch. 6.5 - A bagel shop has onion bagels, poppy seed bagels,...Ch. 6.5 - io. A croissant shop has plain croissants, cherry...Ch. 6.5 - ii. Howmany ways are there to choose eight coins...Ch. 6.5 - Homy different combinations of pennies, nickels,...Ch. 6.5 - Prob. 13ECh. 6.5 - How many solutions are there to the equation -T| +...Ch. 6.5 - How many solutions are there to the equation -T |...Ch. 6.5 - i6. How many solutions are there to the equation...Ch. 6.5 - strings of 10 ternary digits (o, 1. or 2) are...Ch. 6.5 - ,How many strings of 20-decima] digits are there...Ch. 6.5 - Prob. 19ECh. 6.5 - How many solutions are there to the inequality .ii...Ch. 6.5 - i. A Swedish tour guide has devised a clever way...Ch. 6.5 - w many ways can an airplane pilot be scheduled for...Ch. 6.5 - How many ways are there to distribute six...Ch. 6.5 - How many ways are there to distribute 12...Ch. 6.5 - Howmany wavs aiethereto distribute 12...Ch. 6.5 - Prob. 26ECh. 6.5 - How many positive integers less than 1,000,000...Ch. 6.5 - a8. How many positive integers less than 1,000,000...Ch. 6.5 - Prob. 29ECh. 6.5 - Prob. 30ECh. 6.5 - Prob. 31ECh. 6.5 - How many different strings can be made from the...Ch. 6.5 - How many different strings can be made from the...Ch. 6.5 - different strings can be made from the letters...Ch. 6.5 - How many different strings can be made from the...Ch. 6.5 - How many strings idth five or more characters can...Ch. 6.5 - How many strings with seven or more characters can...Ch. 6.5 - How many different bit strings can be formed using...Ch. 6.5 - Prob. 39ECh. 6.5 - Prob. 40ECh. 6.5 - i....Ch. 6.5 - Prob. 42ECh. 6.5 - How many ways are. there to deal hands of seven...Ch. 6.5 - In bridge. the 52 cards of a standard deck are...Ch. 6.5 - How many ways are there to deal hands of five...Ch. 6.5 - , In how many ways can a dozen books be placed on...Ch. 6.5 - How many ways cannbooks be placed on t...Ch. 6.5 - Prob. 48ECh. 6.5 - Prob. 49ECh. 6.5 - Prob. 50ECh. 6.5 - Prob. 51ECh. 6.5 - How many ways are there to distribute five...Ch. 6.5 - Prob. 53ECh. 6.5 - 54-How many ways are there to put five temporary...Ch. 6.5 - Prob. 55ECh. 6.5 - Prob. 56ECh. 6.5 - Prob. 57ECh. 6.5 - 8,Howmany ways are thereto pack eightidentical...Ch. 6.5 - Prob. 59ECh. 6.5 - 6o. How many ways are there to distribute five...Ch. 6.5 - 6i. How many ways are there to distribute five...Ch. 6.5 - Suppose that a basketball league has 32 teams,...Ch. 6.5 - f 63. Suppose that a weapons inspector must...Ch. 6.5 - Howmanv dififerentterms are therein the expansion...Ch. 6.5 - Prob. 65ECh. 6.5 - Prob. 66ECh. 6.5 - Find the coefficient ofi3y2z5 in Qc + y + z)Ch. 6.5 - How many terms are there in the expansionCh. 6.6 - ...Ch. 6.6 - ...Ch. 6.6 - Prob. 3ECh. 6.6 - Prob. 4ECh. 6.6 - Find the next larger permutation in lexicographic...Ch. 6.6 - Find the next larger permutation in lexicographic,...Ch. 6.6 - Use Algorithm 1 to generate the 24 permutations of...Ch. 6.6 - Prob. 8ECh. 6.6 - Use Algorithm 3 to listallthe 3-combinations of{1,...Ch. 6.6 - Show that Algorithm1produces the next larger...Ch. 6.6 - Show that Algorithm 3 produces the next larger...Ch. 6.6 - Develop an algorithm for generating the...Ch. 6.6 - List all 3-permutations of {1,2,3,4,5}. The...Ch. 6.6 - Find the Cantor digits an ti2,that correspond to...Ch. 6.6 - Prob. 15ECh. 6.6 - i6,Find the permutations of {1,2,3,4,5} that...Ch. 6.6 - Prob. 17ECh. 6 - Explain how the sum and product rules can be used...Ch. 6 - Explain how to find the number of bit strings of...Ch. 6 - Prob. 3RQCh. 6 - How can yon find the number of possible outcomes...Ch. 6 - How can you find the number of bit strings...Ch. 6 - State the pigeonhole principle, Explain how the...Ch. 6 - State the generalized pigeonhole principle....Ch. 6 - ft What is the difference between an r-combination...Ch. 6 - What i s Pas cal's tri angle? How can arow of...Ch. 6 - What is meant by a combinatorial proof of an...Ch. 6 - ii. Explain how to prove Pascal's identity using a...Ch. 6 - Stateth e bin omial th eor em. Explain how to pr o...Ch. 6 - Explain how to find a formula for the number of...Ch. 6 - Letnand r be positive integers. Explain why the...Ch. 6 - Prob. 15RQCh. 6 - Prob. 16RQCh. 6 - a) How many ways are there to deal hands of five...Ch. 6 - Describe an algorithm for generating all the...Ch. 6 - i. How many ways are there to choose 6 items from...Ch. 6 - a.H 01 v many ways ar e ther e to ch o o se1o...Ch. 6 - Prob. 3SECh. 6 - How many strings of length10either start with ooo...Ch. 6 - Prob. 5SECh. 6 - Prob. 6SECh. 6 - Prob. 7SECh. 6 - Hoi v many positive integers less than iqoo have...Ch. 6 - Prob. 9SECh. 6 - Prob. 10SECh. 6 - Prob. 11SECh. 6 - How many people are needed to guarantee that at...Ch. 6 - Show that given anv set of 10 positive integers...Ch. 6 - Prob. 14SECh. 6 - Prob. 15SECh. 6 - Prob. 16SECh. 6 - Show that in a sequence ofmintegers there exists...Ch. 6 - Prob. 18SECh. 6 - Show that the decimal expansion of a rational...Ch. 6 - Once a computer worm infects a personal computer...Ch. 6 - si.How many ways are there to choose a dozen...Ch. 6 - ss.Findn if P(n,2] = 110. J\?i, n] = 5040....Ch. 6 - Prob. 23SECh. 6 - Show that ifnandrare nonnegative integers and n >...Ch. 6 - Prob. 25SECh. 6 - Give a combinatorial proof ofCorollary 2ofSection...Ch. 6 - Prob. 27SECh. 6 - a8. Prove using mathematical induction that O>• 2)...Ch. 6 - Prob. 29SECh. 6 - Show that V7' XIt. I = (’) if nis an integer withCh. 6 - Prob. 31SECh. 6 - Prob. 32SECh. 6 - How many bit strings of length n, where n > 4,...Ch. 6 - Prob. 34SECh. 6 - Prob. 35SECh. 6 - Prob. 36SECh. 6 - How many ways are there to assign 24 students to...Ch. 6 - Prob. 38SECh. 6 - - How many solutions are there to the equation xt...Ch. 6 - How many different strings can be made from the...Ch. 6 - How many subsets of a set with ten el e m ents...Ch. 6 - Prob. 42SECh. 6 - Prob. 43SECh. 6 - How many ways are. there to seat six boys and...Ch. 6 - How many ways are there to distribute six objects...Ch. 6 - How many ways are there to distribute five obj...Ch. 6 - Find these signless Stirling numb er s of the...Ch. 6 - Show that ifnis a positive integer, then ,Ch. 6 - Prob. 49SECh. 6 - Prob. 50SECh. 6 - Prob. 51SECh. 6 - j2, How many n-element RXA sequences consist of 4...Ch. 6 - *53. Suppose that when an enzyme that breaks RXA...Ch. 6 - Suppose that when an enzyme that breaks RXA chains...Ch. 6 - Devise an algorithm for generating all the...Ch. 6 - Devise an algorithm for generating all the...Ch. 6 - Prob. 57SECh. 6 - Prob. 58SECh. 6 - Prob. 1CPCh. 6 - Prob. 2CPCh. 6 - Prob. 3CPCh. 6 - Prob. 4CPCh. 6 - Given a positive integern,listallthe permutations...Ch. 6 - Prob. 6CPCh. 6 - Prob. 7CPCh. 6 - Prob. 8CPCh. 6 - Prob. 9CPCh. 6 - Prob. 10CPCh. 6 - Prob. 1CAECh. 6 - Prob. 2CAECh. 6 - Prob. 3CAECh. 6 - Prob. 4CAECh. 6 - Prob. 5CAECh. 6 - Prob. 6CAECh. 6 - Prob. 7CAECh. 6 - Prob. 8CAECh. 6 - Prob. 9CAECh. 6 - Describe some of the earliest uses of the...Ch. 6 - Prob. 2WPCh. 6 - Discuss the importance of combinatorial reasoning...Ch. 6 - Jlanv combinatonal identities are described in...Ch. 6 - Prob. 5WPCh. 6 - Prob. 6WPCh. 6 - Prob. 7WPCh. 6 - Describe the latest discoveries of values and...Ch. 6 - Prob. 9WPCh. 6 - Prob. 10WP
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