Chapter 6 counting

Counting Counting Huynh Tuong Nguyen, Tran Vinh Tan 6 1 Chapter 6 Counting Discrete Structures for Computing on 25 April 2011 Huynh Tuong Nguyen, Tran Vinh Tan Faculty of Computer Science and Engineer[.]

Trang 1

Huynh Tuong Nguyen, Tran Vinh Tan

Chapter 6

Counting

Discrete Structures for Computing on 25 April 2011

Huynh Tuong Nguyen, Tran Vinh TanFaculty of Computer Science and Engineering

University of Technology - VNUHCM

Trang 2

Contents

Trang 4

Introduction

Example

Trang 5

Trang 6

Problems

brute force attack

Trang 7

Problems

brute force attack

Trang 8

Problems

brute force attack

Trang 9

ports How many different ports in the center?

Solution

There are two tasks to choose a port:

Because there are 32 ways to choose the router and 24 ways tochoose the port no matter which router has been selected, thenumber of ports are 32 × 24 = 768 ports

Definition (Product Rule (Luật nhân))

two tasks If there are n1ways to do the first task and for each of

Can be extended to T1, T2, , Tmtasks in sequence

Trang 10

Solution

Because there are 32 ways to choose the router and 24 ways to

choose the port no matter which router has been selected, the

number of ports are 32 × 24 = 768 ports

Trang 11

Solution

Trang 12

Product Rule

Example

Solution

Trang 13

Two new students arrive at the dorm and there are 12 rooms

two students?

Example (2)How many different bit strings of length seven are there?

Example (3)How many one-to-one functions are there from a set with melements to one with n elements?

Trang 14

two students?

Example (2)

How many different bit strings of length seven are there?

Example (3)How many one-to-one functions are there from a set with melements to one with n elements?

Trang 15

More examples

Example (1)

two students?

Example (2)

How many different bit strings of length seven are there?

Example (3)

How many one-to-one functions are there from a set with m

elements to one with n elements?

Trang 16

A student can choose a project from one of three fields:

Information system (32 projects), Software Engineering (12

projects) and Computer Science (15 projects) How many ways are

there for a student to choose?

Solution: 32 + 12 + 15Definition (Sum Rule (Luật cộng))

set of n2 ways, then there aren1+ n2 ways to do the task

Can be extended to n1, n2, , nmdisjoint ways

Trang 17

Solution: 32 + 12 + 15

Definition (Sum Rule (Luật cộng))

Trang 18

Solution: 32 + 12 + 15

Trang 19

Sum Rule

Example

Solution: 32 + 12 + 15

Trang 20

or two alphanumeric characters, where uppercase and lowercase

begin with a letter and must bedifferent fromthe five strings of

two characters that are reserved for programming use How many

different variables names are there in this language?

Solution

Let V equal to the number of different variable names

the number of these that are two characters long Then, by sumrule, V = V1+ V2

are 26 · 36 strings of length two that begin with a letter and endwith an alphanumeric character However, five of these are

different names for variables in this language

Trang 21

Using Both Rules

Example

or two alphanumeric characters, where uppercase and lowercase

begin with a letter and must bedifferent fromthe five strings of

two characters that are reserved for programming use How many

different variables names are there in this language?

Solution

Let V equal to the number of different variable names

the number of these that are two characters long Then, by sum

rule, V = V1+ V2

are 26 · 36 strings of length two that begin with a letter and end

with an alphanumeric character However, five of these are

different names for variables in this language

Trang 22

ways

Trang 23

ways

Trang 24

ways

Trang 25

ways

Trang 26

ways

Trang 27

Inclusion-Exclusion

Example

Solution

ways

Trang 30

Inclusion-Exclusion

|A ∪ B| = |A| + |B| − |A ∩ B|

Trang 34

Inclusion-Exclusion

|A ∪ B ∪ C| =???

Trang 35

Example

onlyArsenal,10liked onlyChelsea,12liked onlyMUand40liked

all three clubs How many of the student surveyed likedboth MU

andChelseabut notArsenal?

Trang 36

Pigeonhole Principle

Trang 37

thesame birthday.

Because there are only 366 possible birthdays

Example (2)

Because there are 26 letters in the English alphabet

Trang 38

thesame birthday.

Example (2)

Trang 39

thesame birthday.

Example (2)

Trang 40

Examples

Example (1)

thesame birthday

Example (2)

Trang 41

Examples

Example (1)

thesame birthday

Example (2)

Trang 42

Prove that if seven distinct numbers are selected from

{1, 2, , 11}, then some two of these numbers sum to 12

Solution

{3, 9}, {4, 8}, {5, 7}, {6}

pigeonhole corresponding to the set that contains it

placed in six pigeonholes, some pigeonhole contains twonumbers

Trang 43

Solution

{3, 9}, {4, 8}, {5, 7}, {6}

Trang 44

Solution

{3, 9}, {4, 8}, {5, 7}, {6}

Trang 45

Solution

{3, 9}, {4, 8}, {5, 7}, {6}

Trang 46

Exercise

Example

Solution

{3, 9}, {4, 8}, {5, 7}, {6}

placed in six pigeonholes, some pigeonhole contains two

Trang 49

Examples – Permutations

How many ways can we arrange three students to stand in line for

a picture?

Number of choices: 6 = 3!

Trang 50

Apermutation(hoán vị) of a set of distinct objects is anordered

arrangementof these objects

An ordered arrangement of r elements of a set is called an

r-permutation (hoán vị chập r)

(n − r)!

Example

second-prizewinner, and athird-prizewinner from 100 differentpeople who have entered a contest?

P (100, 3) = 100 · 99 · 98 = 970, 200

Trang 51

(n − r)!

Example

second-prizewinner, and athird-prizewinner from 100 different

people who have entered a contest?

P (100, 3) = 100 · 99 · 98 = 970, 200

Trang 52

Permutations

Definition

(n − r)!

Example

second-prizewinner, and athird-prizewinner from 100 different

people who have entered a contest?

Trang 55

Examples – Combinations

How many ways to choose two students from a group of four to

offer scholarship?

Number of choices: 6

Trang 56

Anr-combination(tổ hợp chập r) of elements of a set is an

unordered selectionof r elements from the set Thus, an

r-combination is simply a subset of the set with r elements

Trang 57

How many ways are there to select eleven players from a

22-member football team to start up?

Trang 58

Combinations

Definition (Combinations)

How many ways are there to select eleven players from a

22-member football team to start up?

22!

Trang 59

Exercises – Permutations with Repetition

must begin her trip in a specified city, but she can visit the

other seven cities in any order she wishes How many possible

orders can the salesman use when visiting these cities?

and 11 in CE department How many ways are there to select

a defend committee if the committee is to consist of three

faculty members from the CS and four from the CE

department?

Trang 62

Permutations with Repetition

Trang 63

Example

Question: How many ways we can choose3 students fromthe

facultiesof Computer Science, Electrical Engineering and

Mechanical Engineering?

Trang 64

Trang 66

Trang 67

There are C(n + r − 1, r) r-combinations from a set with n

elements when repetition of elements is allowed

ExampleHow many solutions does the equation

x1+ x2+ x3= 11have, where x1, x2, and x3are nonnegative integers?

Trang 68

Combinations with Repetition

Theorem

There are C(n + r − 1, r) r-combinations from a set with n

elements when repetition of elements is allowed

Example

How many solutions does the equation

x1+ x2+ x3= 11have, where x1, x2, and x3are nonnegative integers?

Trang 69

Examples

Question: How many permutations are there ofMISSISSIPPI?

Trang 70

Examples

Question: How many permutations are there ofMISSISSIPPI?

Trang 71

The number of different permutations of n objects, where there

are n1indistinguishable objects of type 1, n2 indistinguishable

objects of type 2, , and nkindistinguishable objects of type k, is

n!

n1!n2! · · · nk!

ExampleHow many permutations are there of MISSISSIPPI?

Trang 72

Permutations with Indistinguishable Objects

Theorem

The number of different permutations of n objects, where there

are n1indistinguishable objects of type 1, n2 indistinguishable

objects of type 2, , and nkindistinguishable objects of type k, is

Tiêu đề	Counting
Tác giả	Huynh Tuong Nguyen, Tran Vinh Tan
Trường học	University of Technology - VNUHCM
Chuyên ngành	Computer Science and Engineering
Thể loại	Chương
Năm xuất bản	2011
Thành phố	Ho Chi Minh City

Định dạng
Số trang	72
Dung lượng	563,44 KB