You are on page 1of 2

# Combinatorics, the study of arrangements of objects, is an important part of dis

crete mathematics. This subject was studied as long ago as the seventeenth century, when c
ombinatorial questions arose in the study of gambling games. Enumeration, the counti
ng of objects
with certain properties, is an important part of combinatorics. We must count ob
jects to solve
many different types of problems. For instance, counting is used to determine th
e complexity of
algorithms. Counting is also required to determine whether there are enough tele
phone numbers
or Internet protocol addresses tomeet demand. Recently, it has played a key role
inmathematical
biology, especially in sequencing DNA. Furthermore, counting techniques are used
extensively
when probabilities of events are computed.
The basic rules of counting, which we will study in Section 6.1, can solve a tre
mendous
variety of problems. For instance, we can use these rules to enumerate the diffe
rent telephone
numbers possible in the United States, the allowable passwords on a computer sys
tem, and the
different orders in which the runners in a race can ?nish. Another important com
binatorial tool
is the pigeonhole principle, which we will study in Section 6.2. This states tha
t when objects are
placed in boxes and there are more objects than boxes, then there is a box conta
ining at least two objects. For instance, we can use this principle to show that
among a set of 15 or more students,
at least 3 were born on the same day of the week.
We can phrase many counting problems in terms of ordered or unordered arrangemen
ts of
the objects of a set with or without repetitions. These arrangements, called per
mutations and
combinations, are used in many counting problems. For instance, suppose the 100
top ?nishers
on a competitive examtaken by 2000 students are invited to a banquet.We can coun
t the possible
sets of 100 students that will be invited, as well as the ways in which the top
10 prizes can be
awarded.
Another problem in combinatorics involves generating all the arrangements of a s
peci?ed
kind. This is often important in computer simulations. We will devise algorithms
to generate
arrangements of various types.
The Basics of Counting
Introduction
Suppose that a password on a computer system consists of six, seven, or eight ch
aracters. Each
of these charactersmust be a digit or a letter of the alphabet. Each passwordmus
t contain at least
one digit. How many such passwords are there? The techniques needed to answer th
is question
and a wide variety of other counting problems will be introduced in this section
.
Counting problems arise throughout mathematics and computer science. For example

, we
must count the successful outcomes of experiments and all the possible outcomes
of these
experiments to determine probabilities of discrete events. We need to count the
number of
operations used by an algorithm to study its time complexity.
We will introduce the basic techniques of counting in this section. These method
s serve as
the foundation for almost all counting techniques.
Basic Counting Principles
We ?rst present two basic counting principles, the product rule and the sumrule.
Then we will
show how they can be used to solve many different counting problems.
The product rule applies when a procedure is made up of separate tasks.
THE PRODUCT RULE Suppose that a procedure can be broken down into a sequence of
two tasks. If there are n1 ways to do the ?rst task and for each of these ways o
f doing the ?rst
task, there are n2 ways to do the second task, then there are n1n2 ways to do th
e procedure.
Examples 1 10 show how the product rule is used.
A new company with just two employees, Sanchez and Patel, rents a ?oor of a buil
ding with
12 of?ces. How many ways are there to assign different of?ces to these two emplo
yees?
Solution: The procedure of assigning of?ces to these two employees consists of a
ssigning an
of?ce to Sanchez, which can be done in 12 ways, then assigning an of?ce to Patel
different from
the of?ce assigned to Sanchez, which can be done in 11 ways. By the product rule
, there are
= ?
The text passage is an excerpt from a textbook on combinatorics, discussing the importance of counting and basic counting principles, including the product rule. It includes a specific example of applying the product rule to assign offices to two employees.
single_question_with_answer

the passage contains a single mathematical question with an answer provided. the question is about the number of ways to assign different offices to two employees, and the answer is calculated using the product rule.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
You are on page 1of 2

# Combinatorics, the study of arrangements of objects, is an important part of dis

crete mathematics. This subject was studied as long ago as the seventeenth century, when c
ombinatorial questions arose in the study of gambling games. Enumeration, the counti
ng of objects
with certain properties, is an important part of combinatorics. We must count ob
jects to solve
many different types of problems. For instance, counting is used to determine th
e complexity of
algorithms. Counting is also required to determine whether there are enough tele
phone numbers
or Internet protocol addresses tomeet demand. Recently, it has played a key role
inmathematical
biology, especially in sequencing DNA. Furthermore, counting techniques are used
extensively
when probabilities of events are computed.
The basic rules of counting, which we will study in Section 6.1, can solve a tre
mendous
variety of problems. For instance, we can use these rules to enumerate the diffe
rent telephone
numbers possible in the United States, the allowable passwords on a computer sys
tem, and the
different orders in which the runners in a race can ?nish. Another important com
binatorial tool
is the pigeonhole principle, which we will study in Section 6.2. This states tha
t when objects are
placed in boxes and there are more objects than boxes, then there is a box conta
ining at least two objects. For instance, we can use this principle to show that
among a set of 15 or more students,
at least 3 were born on the same day of the week.
We can phrase many counting problems in terms of ordered or unordered arrangemen
ts of
the objects of a set with or without repetitions. These arrangements, called per
mutations and
combinations, are used in many counting problems. For instance, suppose the 100
top ?nishers
on a competitive examtaken by 2000 students are invited to a banquet.We can coun
t the possible
sets of 100 students that will be invited, as well as the ways in which the top
10 prizes can be
awarded.
Another problem in combinatorics involves generating all the arrangements of a s
peci?ed
kind. This is often important in computer simulations. We will devise algorithms
to generate
arrangements of various types.
The Basics of Counting
Introduction
Suppose that a password on a computer system consists of six, seven, or eight ch
aracters. Each
of these charactersmust be a digit or a letter of the alphabet. Each passwordmus
t contain at least
one digit. How many such passwords are there? The techniques needed to answer th
is question
and a wide variety of other counting problems will be introduced in this section
.
Counting problems arise throughout mathematics and computer science. For example

, we
must count the successful outcomes of experiments and all the possible outcomes
of these
experiments to determine probabilities of discrete events. We need to count the
number of
operations used by an algorithm to study its time complexity.
We will introduce the basic techniques of counting in this section. These method
s serve as
the foundation for almost all counting techniques.
Basic Counting Principles
We ?rst present two basic counting principles, the product rule and the sumrule.
Then we will
show how they can be used to solve many different counting problems.
The product rule applies when a procedure is made up of separate tasks.
THE PRODUCT RULE Suppose that a procedure can be broken down into a sequence of
two tasks. If there are n1 ways to do the ?rst task and for each of these ways o
f doing the ?rst
task, there are n2 ways to do the second task, then there are n1n2 ways to do th
e procedure.
Examples 1 10 show how the product rule is used.
A new company with just two employees, Sanchez and Patel, rents a ?oor of a buil
ding with
12 of?ces. How many ways are there to assign different of?ces to these two emplo
yees?
Solution: The procedure of assigning of?ces to these two employees consists of a
ssigning an
of?ce to Sanchez, which can be done in 12 ways, then assigning an of?ce to Patel
different from
the of?ce assigned to Sanchez, which can be done in 11 ways. By the product rule
, there are
= ?
user
The text passage is an excerpt from a textbook on combinatorics, discussing the importance of counting and basic counting principles, including the product rule. It includes a specific example of applying the product rule to assign offices to two employees.

TYPE: SINGLE_QUESTION_WITH_ANSWER

The passage contains a single mathematical question with an answer provided. The question is about the number of ways to assign different offices to two employees, and the answer is calculated using the product rule.