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Chapter 6

Counting

Educators


Problem 1

There are 18 mathematics majors and 325 computer science majors at a college.
a) In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major?
b) In how many ways can one representative be picked who is either a mathematics major or a computer science major?

Clarissa N.
Numerade Educator

Problem 2

An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?

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Problem 3

A multiple-choice test contains 10 questions. There are four possible answers for each question.
a) In how many ways can a student answer the questions on the test if the student answers every question?
b) In how many ways can a student answer the questions on the test if the student can leave answers blank?

Clarissa N.
Numerade Educator

Problem 4

A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?

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Problem 5

Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from New York to San Francisco via Denver, when you pick an airline for the flight to Denver and an airline for the continuation flight to San Francisco?

Clarissa N.
Numerade Educator

Problem 6

There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit?

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Problem 7

How many different three-letter initials can people have?

Clarissa N.
Numerade Educator

Problem 8

How many different three-letter initials with none of the letters repeated can people have?

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Problem 9

How many different three-letter initials are there that begin with an A?

Clarissa N.
Numerade Educator

Problem 10

How many bit strings are there of length eight?

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Problem 11

How many bit strings of length ten both begin and end with a 1?

Clarissa N.
Numerade Educator

Problem 12

How many bit strings are there of length six or less, not counting the empty string?

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Problem 13

How many bit strings with length not exceeding n, where n is a positive integer, consist entirely of 1s, not counting the empty string?

Clarissa N.
Numerade Educator

Problem 14

How many bit strings of length n, where n is a positive integer, start and end with 1s?

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Problem 15

How many strings are there of lowercase letters of length four or less, not counting the empty string?

Clarissa N.
Numerade Educator

Problem 16

How many strings are there of four lowercase letters that have the letter x in them?

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Problem 17

How many strings of five ASCII characters contain the character @ (“at” sign) at least once? [Note: There are 128 different ASCII characters.]

Clarissa N.
Numerade Educator

Problem 18

How many 5-element DNA sequences
a) end with A?
b) start with T and end with G?
c) contain only A and T?
d) do not contain C?

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Problem 19

How many 6-element RNA sequences
a) do not contain U?
b) end with GU?
c) start with C?
d) contain only A or U?

Clarissa N.
Numerade Educator

Problem 20

How many positive integers between 5 and 31
a) are divisible by 3? Which integers are these?
b) are divisible by 4? Which integers are these?
c) are divisible by 3 and by 4? Which integers are these?

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Problem 21

How many positive integers between 50 and 100
a) are divisible by 7? Which integers are these?
b) are divisible by 11? Which integers are these?
c) are divisible by both 7 and 11? Which integers are these?

Clarissa N.
Numerade Educator

Problem 22

How many positive integers less than 1000
a) are divisible by 7?
b) are divisible by 7 but not by 11?
c) are divisible by both 7 and 11?
d) are divisible by either 7 or 11?
e) are divisible by exactly one of 7 and 11?
f ) are divisible by neither 7 nor 11?
g) have distinct digits?
h) have distinct digits and are even?

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Problem 23

How many positive integers between 100 and 999 inclusive
a) are divisible by 7?
b) are odd?
c) have the same three decimal digits?
d) are not divisible by 4?
e) are divisible by 3 or 4?
f ) are not divisible by either 3 or 4?
g) are divisible by 3 but not by 4?
h) are divisible by 3 and 4?

Clarissa N.
Numerade Educator

Problem 24

How many positive integers between 1000 and 9999 inclusive
a) are divisible by 9?
b) are even?
c) have distinct digits?
d) are not divisible by 3?
e) are divisible by 5 or 7?
f ) are not divisible by either 5 or 7?
g) are divisible by 5 but not by 7?
h) are divisible by 5 and 7?

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Problem 25

How many strings of three decimal digits
a) do not contain the same digit three times?
b) begin with an odd digit?
c) have exactly two digits that are 4s?

Clarissa N.
Numerade Educator

Problem 26

How many strings of four decimal digits
a) do not contain the same digit twice?
b) end with an even digit?
c) have exactly three digits that are 9s?

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Problem 27

A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from that state. How many ways are there to form this committee?

Clarissa N.
Numerade Educator

Problem 28

How many license plates can be made using either three digits followed by three uppercase English letters or three uppercase English letters followed by three digits?

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Problem 29

How many license plates can be made using either two uppercase English letters followed by four digits or two digits followed by four uppercase English letters?

Clarissa N.
Numerade Educator

Problem 30

How many license plates can be made using either three uppercase English letters followed by three digits or four uppercase English letters followed by two digits?

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Problem 31

How many license plates can be made using either two or three uppercase English letters followed by either two or three digits?

Clarissa N.
Numerade Educator

Problem 32

How many strings of eight uppercase English letters are there
a) if letters can be repeated?
b) if no letter can be repeated?
c) that start with X, if letters can be repeated?
d) that start with X, if no letter can be repeated?
e) that start and end with X, if letters can be repeated?
f ) that start with the letters BO (in that order), if letters can be repeated?
g) that start and end with the letters BO (in that order), if letters can be repeated?
h) that start or end with the letters BO (in that order), if letters can be repeated?

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Problem 33

How many strings of eight English letters are there
a) that contain no vowels, if letters can be repeated?
b) that contain no vowels, if letters cannot be repeated?
c) that start with a vowel, if letters can be repeated?
d) that start with a vowel, if letters cannot be repeated?
e) that contain at least one vowel, if letters can be repeated?
f ) that contain exactly one vowel, if letters can be repeated?
g) that start with X and contain at least one vowel, if letters can be repeated?
h) that start and end with X and contain at least one vowel, if letters can be repeated?

Clarissa N.
Numerade Educator

Problem 34

How many different functions are there from a set with 10 elements to sets with the following numbers of elements?
a) 2 b) 3 c) 4 d) 5

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Problem 35

How many one-to-one functions are there from a set with five elements to sets with the following number of elements?
a) 4 $\quad$ b) 5 $\quad$ c) 6 $\quad$ d) 7

Clarissa N.
Numerade Educator

Problem 36

How many functions are there from the set ${1, 2, ... , n},$ where n is a positive integer, to the set ${0, 1}$?

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Problem 37

How many functions are there from the set ${1, 2, ... , n},$ where n is a positive integer, to the set ${0, 1}$
a) that are one-to-one?
b) that assign 0 to both 1 and n?
c) that assign 1 to exactly one of the positive integers less than n?

Clarissa N.
Numerade Educator

Problem 38

How many partial functions (see Section 2.3) are there from a set with five elements to sets with each of these number of elements?
a) 1 $\quad$ b) 2 $\quad$ c) 5 $\quad$ d) 9

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Problem 39

How many partial functions (see Definition 13 of Section 2.3) are there from a set with m elements to a set with n elements, where m and n are positive integers?

Clarissa N.
Numerade Educator

Problem 40

How many subsets of a set with 100 elements have more than one element?

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Problem 41

A palindrome is a string whose reversal is identical to the string. How many bit strings of length n are palindromes?

Clarissa N.
Numerade Educator

Problem 42

How many 4-element DNA sequences
a) do not contain the base T?
b) contain the sequence ACG?
c) contain all four bases A, T, C, and G?
d) contain exactly three of the four bases A, T, C, and G?

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Problem 43

How many 4-element RNA sequences
a) contain the base U?
b) do not contain the sequence CUG?
c) do not contain all four bases A, U, C, and G?
d) contain exactly two of the four bases A, U, C, and G?

Clarissa N.
Numerade Educator

Problem 44

On each of the 22 work days in a particular month, every employee of a start-up venture was sent a company communication. If a total of 4642 total company communications were sent, how many employees does the company have, assuming that no staffing changes were made that month?

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Problem 45

At a large university, 434 freshmen, 883 sophomores, and 43 juniors are enrolled in an introductory algorithms course. How many sections of this course need to be scheduled to accommodate all these students if each section contains 34 students?

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Problem 46

How many ways are there to seat four of a group of ten people around a circular table where two seatings are considered the same when everyone has the same immediate left and immediate right neighbor?

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Problem 47

How many ways are there to seat four of a group of ten people around a circular table where two seatings are considered the same when everyone has the same immediate left and immediate right neighbor?

Clarissa N.
Numerade Educator

Problem 48

In how many ways can a photographer at a wedding arrange 6 people in a row from a group of 10 people, where the bride and the groom are among these 10 people, if
a) the bride must be in the picture?
b) both the bride and groom must be in the picture?
c) exactly one of the bride and the groom is in the picture?

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Problem 49

In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if
a) the bride must be next to the groom?
b) the bride is not next to the groom?
c) the bride is positioned somewhere to the left of the groom?

Clarissa N.
Numerade Educator

Problem 50

How many bit strings of length seven either begin with two 0s or end with three 1s?

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Problem 51

How many bit strings of length 10 either begin with three 0s or end with two 0 $\mathrm{s} ?$

Clarissa N.
Numerade Educator

Problem 52

How many bit strings of length 10 contain either five consecutive 0 s or five consecutive 1 $\mathrm{s} ?$

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Problem 53

How many bit strings of length eight contain either three consecutive 0 s or four consecutive 1 $\mathrm{s} ?$

Clarissa N.
Numerade Educator

Problem 54

Every student in a discrete mathematics class is either a computer science or a mathematics major or is a joint major in these two subjects. How many students are in the class if there are 38 computer science majors (including joint majors), 23 mathematics majors (including joint majors), and 7 joint majors?

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Problem 55

How many positive integers not exceeding 100 are divisible either by 4 or by 6?

Clarissa N.
Numerade Educator

Problem 56

How many different initials can someone have if a person has at least two, but no more than five, different initials? Assume that each initial is one of the 26 uppercase letters of the English language.

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Problem 57

Suppose that a password for a computer system must have at least 8, but no more than 12, characters, where each character in the password is a lowercase English letter, an uppercase English letter, a digit, or one of the six special characters $?, >, <, !, +,$ and $=.$
a) How many different passwords are available for this computer system?
b) How many of these passwords contain at least one occurrence of at least one of the six special characters?
c) Using your answer to part (a), determine how long it takes a hacker to try every possible password, assuming that it takes one nanosecond for a hacker to check each possible password.

Clarissa N.
Numerade Educator

Problem 58

The name of a variable in the C programming language is a string that can contain uppercase letters, lowercase letters, digits, or underscores. Further, the first character in the string must be a letter, either uppercase or lowercase, or an underscore. If the name of a variable is determined by its first eight characters, how many different variables can be named in C? (Note that the name of a variable may contain fewer than eight characters.)

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Problem 59

The name of a variable in the JAVA programming language is a string of between 1 and 65,535 characters, inclusive, where each character can be an uppercase or a lowercase letter, a dollar sign, an underscore, or a digit, except that the first character must not be a digit. Determine the number of different variable names in JAVA.

Clarissa N.
Numerade Educator

Problem 60

The International Telecommunications Union (ITU) specifies that a telephone number must consist of a country code with between 1 and 3 digits, except that the code 0 is not available for use as a country code, followed by a number with at most 15 digits. How many available possible telephone numbers are there that satisfy these restrictions?

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Problem 61

Suppose that at some future time every telephone in the world is assigned a number that contains a country code 1 to 3 digits long, that is, of the form $X, XX,$ or $XXX,$ followed by a 10-digit telephone number of the form $NXX-NXX-XXXX$ (as described in Example 8). How many different telephone numbers would be available worldwide under this numbering plan?

Clarissa N.
Numerade Educator

Problem 62

A key in the Vigenère cryptosystem is a string of English letters, where the case of the letters does not matter. How many different keys for this cryptosystem are there with three, four, five, or six letters?

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Problem 63

A wired equivalent privacy (WEP) key for a wireless fidelity (WiFi) network is a string of either $10,26,$ or 58 hexadecimal digits. How many different WEP keys are there?

Clarissa N.
Numerade Educator

Problem 64

Suppose that $p$ and $q$ are prime numbers and that $n=p q$ . Use the principle of inclusion-exclusion to find the number of positive integers not exceeding $n$ that are relatively prime to $n .$

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Problem 65

Use the principle of inclusion-exclusion to find the number of positive integers less than $1,000,000$ that are not divisible by either 4 or by 6 .

Clarissa N.
Numerade Educator

Problem 66

Use a tree diagram to find the number of bit strings of length four with no three consecutive 0 s.

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Problem 67

How many ways are there to arrange the letters $a, b, c,$ and $d$ such that $a$ is not followed immediately by $b ?$

Clarissa N.
Numerade Educator

Problem 68

Use a tree diagram to find the number of ways that the World Series can occur, where the first team that wins four games out of seven wins the series.

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Problem 69

Use a tree diagram to determine the number of subsets of $\{3,7,9,11,24\}$ with the property that the sum of the elements in the subset is less than 28 .

Clarissa N.
Numerade Educator

Problem 70

a) Suppose that a store sells six varieties of soft drinks: cola, ginger ale, orange, root beer, lemonade, and cream soda. Use a tree diagram to determine the number of different types of bottles the store must stock to have all varieties available in all size bottles if all varieties are available in 12-ounce bottles, all but lemonade are available in 20-ounce bottles, only cola and ginger ale are available in 32-ounce bottles, and all but lemonade and cream soda are available in 64-ounce bottles?
b) Answer the question in part (a) using counting rules.

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Problem 71

a) Suppose that a popular style of running shoe is available for both men and women. The woman’s shoe comes in sizes 6, 7, 8, and 9, and the man’s shoe comes in sizes 8, 9, 10, 11, and 12. The man’s shoe comes in white and black, while the woman’s shoe comes in white, red, and black. Use a tree diagram to determine the number of different shoes that a store has to stock to have at least one pair of this type of running shoe for all available sizes and colors for both men and women.
b) Answer the question in part (a) using counting rules.

Clarissa N.
Numerade Educator

Problem 72

Determine the number of matches played in a single-elimination tournament with n players, where for each game between two players the winner goes on, but the loser is eliminated.

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Problem 73

Determine the minimum and the maximum number of matches that can be played in a double-elimination tournament with n players, where after each game between two players, the winner goes on and the loser goes on if and only if this is not a second loss.

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Problem 74

Use the product rule to show that there are $2^{2 n}$ different truth tables for propositions in $n$ variables.

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Problem 75

Use mathematical induction to prove the sum rule for m tasks from the sum rule for two tasks.

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Problem 76

Use mathematical induction to prove the product rule for m tasks from the product rule for two tasks.

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Problem 77

How many diagonals does a convex polygon with n sides have? (Recall that a polygon is convex if every line segment connecting two points in the interior or boundary of the polygon lies entirely within this set and that a diagonal of a polygon is a line segment connecting two vertices that are not adjacent.)

Clarissa N.
Numerade Educator

Problem 78

Data are transmitted over the Internet in datagrams, which are structured blocks of bits. Each datagram contains header information organized into a maximum of 14 different fields (specifying many things, including the source and destination addresses) and a data area that contains the actual data that are transmitted. One of the 14 header fields is the header length field (denoted by HLEN), which is specified by the protocol to be 4 bits long and that specifies the header length in terms of 32-bit blocks of bits. For example, if HLEN = 0110, the header is made up of six 32-bit blocks. Another of the 14 header fields is the 16-bit-long total length field (denoted by TOTAL LENGTH), which specifies the length in bits of the entire datagram, including both the header fields and the data area. The length of the data area is the total length of the datagram minus the length of the header.
a) The largest possible value of TOTAL LENGTH (which is 16 bits long) determines the maximum
total length in octets (blocks of 8 bits) of an Internet datagram. What is this value?
b) The largest possible value of HLEN (which is 4 bits long) determines the maximum total header length in 32-bit blocks. What is this value? What is the maximum total header length in octets?
c) The minimum (and most common) header length is 20 octets. What is the maximum total length in octets of the data area of an Internet datagram?
d) How many different strings of octets in the data area can be transmitted if the header length is 20 octets and the total length is as long as possible?

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