02.25.09 Discrete Mathematics. Exercises Exercise 1:  There are 18 Computer Science (CS) majors and 325 Business Administration (BA) majors at a college.

Slides:

Advertisements

Similar presentations

The Pigeonhole Principle College of Information Technology & Design

Advertisements

Practice Quiz 3 Recursive Definitions Relations Basic Counting Pigeonhole Principle Permutations & Combinations Discrete Probability.

Counting Chapter 6 With Question/Answer Animations.

Application: The Pigeonhole Principle Lecture 37 Section 7.3 Wed, Apr 4, 2007.

Recursively Defined Functions

The Pigeonhole Principle

1 Combinatorics Rosen 6 th ed., § , § Combinatorics Count the number of ways to put things together into various combinations.Count the number.

Permutations and Combinations

PIGEONHOLE PRINCIPLE. Socks You have a drawer full of black and white socks. Without looking in the drawer, how many socks must you pull out to be sure.

Discrete Structures Chapter 5 Pigeonhole Principle Nurul Amelina Nasharuddin Multimedia Department.

The Pigeonhole Principle

Recursive Definitions Rosen, 3.4 Recursive (or inductive) Definitions Sometimes easier to define an object in terms of itself. This process is called.

CSE115/ENGR160 Discrete Mathematics 04/10/12

Combinations We should use permutation where order matters

Recursive Definitions Rosen, 3.4 Recursive (or inductive) Definitions Sometimes easier to define an object in terms of itself. This process is called.

1 The Pigeonhole Principle CS/APMA 202 Rosen section 4.2 Aaron Bloomfield.

CS100 : Discrete Structures

Fall 2015 COMP 2300 Discrete Structures for Computation

Counting. Why counting  Determine the complexity of algorithms To sort n numbers, how many instructions are executed ?  Count the number of objects.

The Basics of Counting Section 6.1.

Counting. Product Rule Example Sum Rule Pigeonhole principle If there are more pigeons than pigeonholes, then there must be at least one pigeonhole.

Chapter The Basics of Counting 5.2 The Pigeonhole Principle

Generalized Permutations and Combinations

Chapter 6 With Question/Answer Animations 1. Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations Binomial Coefficients.

Fall 2002CMSC Discrete Structures1 One, two, three, we’re… Counting.

September1999 CMSC 203 / 0201 Fall 2002 Week #8 – 14/16 October 2002 Prof. Marie desJardins.

Dr. Eng. Farag Elnagahy Office Phone: King ABDUL AZIZ University Faculty Of Computing and Information Technology CPCS 222.

ICS 253: Discrete Structures I Counting and Applications King Fahd University of Petroleum & Minerals Information & Computer Science Department.

The Principle of Inclusion-Exclusion

Module #15: Combinatorics Rosen 5 th ed., §§ & §6.5 Now we are moving on to Ch. 4 It is the study of arrangement of objects e.g. enumeration, counting.

Ch Counting Principles. Example 1  Eight pieces of paper are numbered from 1-8 and placed in a box. One piece of paper is drawn from the box, its.

CSNB143 – Discrete Structure

The Pigeonhole Principle. The pigeonhole principle Suppose a flock of pigeons fly into a set of pigeonholes to roost If there are more pigeons than pigeonholes,

CS 103 Discrete Structures Lecture 16

5.2 The Pigeonhole Principle

Discrete Structures Counting (Ch. 6)

8.6 Counting Principles. Listing Possibilities: Ex 1 Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from.

1 Chapter 2 Pigeonhole Principle. 2 Summary Pigeonhole principle –simple form Pigeonhole principle –strong form Ramsey’s theorem.

5.5 Generalized Permutations and Combinations

1 The Pigeonhole Principle CS 202 Epp section 7.3.

2/24/20161 One, two, three, we’re… Counting. 2/24/20162 Basic Counting Principles Counting problems are of the following kind: “How many different 8-letter.

Introduction Suppose that a password on a computer system consists of 6, 7, or 8 characters. Each of these characters must be a digit or a letter of the.

1 The Pigeonhole Principle CS 202 Epp section ??? Aaron Bloomfield.

CS 104: Discrete Mathematics

The Pigeonhole Principle. Pigeonhole principle The pigeonhole principle : If k is a positive integer and k+1 or more objects are placed into k boxes,

Pigeonhole Principle – Page 1CPTR311 – Discrete Structures CPTR311 Discrete Structures Pigeonhole Principle Reading: Kolman, Section 3.3.

Pigeonhole Principle. If n pigeons fly into m pigeonholes and n > m, then at least one hole must contain two or more pigeons A function from one finite.

Section The Pigeonhole Principle If a flock of 20 pigeons roosts in a set of 19 pigeonholes, one of the pigeonholes must have more than 1 pigeon.

ICS 253: Discrete Structures I Counting and Applications King Fahd University of Petroleum & Minerals Information & Computer Science Department.

Discrete Mathematics and Its Applications Sixth Edition By Kenneth Rosen Chapter 5 Counting 歐亞書局.

Section The Product Rule  Example: How many different license plates can be made if each plate contains a sequence of three uppercase English letters.

COUNTING Discrete Math Team KS MATEMATIKA DISKRIT (DISCRETE MATHEMATICS ) 1.

Section 6.3. Section Summary Permutations Combinations.

Example A standard deck of 52 cards has 13 kinds of cards, with four cards of each of kind, one in each of the four suits, hearts, diamonds, spades, and.

The Pigeonhole Principle

ICS 253: Discrete Structures I

The Pigeonhole Principle

Discrete Structures for Computer Science

Section 16 Inclusion/Exclusion

COCS DISCRETE STRUCTURES

The Pigeonhole Principle

CS 2210 Discrete Structures Counting

Xuan Guo Lab 9 Xuan Guo

CS100: Discrete structures

The Pigeonhole Principle

The Pigeonhole (Dirichlet’s box) Principle

Counting techniques Basic Counting Principles, Pigeonhole Principle, Permutations and Combinations.

The Pigeonhole Principle

Discrete Mathematics and its Applications

Presentation transcript:

Discrete Mathematics

Exercises Exercise 1:  There are 18 Computer Science (CS) majors and 325 Business Administration (BA) majors at a college  How many ways are there to pick two representatives, so that one is a CS major and the other is a BA major?  How many ways are there to pick one representative who is either a CS major or a BA major?

Exercises Exercise 2:  How many different three-letter initials can people have?  How many different three-letter initials with none of the letters repeated can people have?  How many different three-letter initials are there that begin with an A?

Exercises Exercise 3:  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?

Exercises Exercise 3:  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?

Exercises Exercise 4:  Suppose that there are 1807 freshmen at a college. Of these, 453 are taking a course in Computer Science (CS), 567 are taking a course in Business Administration (BA), and 299 are taking courses in both CS and BA. How many are not taking a course either in CS or BA?

Today’s Topics Pigeonhole Principle Permutation

Pigeonhole Principle The Pigeonhole Principle states that, given two natural numbers n and m with n > m, if n items are put into m pigeonholes, then at least one pigeonhole must contain more than one item.

Pigeonhole Principle Examples:  Among any group of 367 people, there must be at least two with the same birthday, because there are only 366 possible birthdays.  In any group of 27 English words, there must be at least two that begin with the same letter, since there are 26 letters in the English alphabet.

Pigeonhole Principle Exercise 1:  How many students must be in a class to guarantee that at least two students receive the same score on the final exam, if the exam is graded on a scale from 0 to 100 points?

Pigeonhole Principle Exercise 2:  Assume that in a box there are 10 black socks and 12 blue socks and you need to get one pair of socks of the same color. Supposing you can take socks out of the box only once and only without looking, what is the minimum number of socks you'd have to pull out at the same time in order to guarantee a pair of the same color?

Pigeonhole Principle The Generalized Pigeonhole Principle states that if N objects are placed into k boxes, then there is at least one box containing at least [N/k] objects. Example:  Among 100 people there are at least [100/12] = 9 who were born in the same month.

Pigeonhole Principle Exercise 3:  What is the minimum number of students required in a discrete math class to be sure that at least six will receive the same grade, if there are 5 possible grades, A, B, C, D, and F?

Pigeonhole Principle Exercise 4:  How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen?

Permutation  An ordered arrangement of the elements of a set r-Permutation  An ordered arrangement of r elements of a set P(n,r)  The number of r-permutations of a set with n elements

Permutation Example 1:  How many ways are there to select a first-prize winner, second-prize winner, and a third-prize winner from 100 different people who have entered a contest? Example 2:  Let S = {1, 2, 3, 4, 5}  What is the 3-permutations of S?