Discrete Structures Counting.

Slides:

Advertisements

Similar presentations

4-1 Chapter 4 Counting Techniques.

Advertisements

1 Counting Techniques: Possibility Trees, Multiplication Rule, Permutations.

College of Information Technology & Design

Counting and Probability The outcome of a random process is sure to occur, but impossible to predict. Examples: fair coin tossing, rolling a pair of dice,

Multiplication Rule. A tree structure is a useful tool for keeping systematic track of all possibilities in situations in which events happen in order.

Discrete Structures Chapter 4 Counting and Probability Nurul Amelina Nasharuddin Multimedia Department.

Discrete Mathematics Lecture 7 More Probability and Counting Harper Langston New York University.

Combinations We should use permutation where order matters

Discrete Mathematics Lecture 6 Alexander Bukharovich New York University.

1 More Counting Techniques Possibility trees Multiplication rule Permutations Combinations.

Copyright © Cengage Learning. All rights reserved. CHAPTER 9 COUNTING AND PROBABILITY.

Lecture 07 Prof. Dr. M. Junaid Mughal

4. Counting 4.1 The Basic of Counting Basic Counting Principles Example 1 suppose that either a member of the faculty or a student in the department is.

Chapter The Basics of Counting 5.2 The Pigeonhole Principle

1 Melikyan/DM/Fall09 Discrete Mathematics Ch. 6 Counting and Probability Instructor: Hayk Melikyan Today we will review sections 6.4,

Advanced Mathematics Counting Techniques. Addition Rule Events (tasks) A and B are mutually exclusive (no common elements/outcomes) and n(A) = a, n(B)

Topics to be covered: Produce all combinations and permutations of sets. Calculate the number of combinations and permutations of sets of m items taken.

9.3 Addition Rule. The basic rule underlying the calculation of the number of elements in a union or difference or intersection is the addition rule.

Methods of Counting Outcomes BUSA 2100, Section 4.1.

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

Chapter 10 – Data Analysis and Probability

1 Lecture 4 (part 1) Combinatorics Reading: Epp Chp 6.

© The McGraw-Hill Companies, Inc., Chapter 4 Counting Techniques.

Lesson Counting Techniques. Objectives Solve counting problems using the Multiplication Rule Solve counting problems using permutations Solve counting.

Part 2 – Factorial and other Counting Rules

Objectives Solve counting problems using the Multiplication Rule Solve counting problems using permutations Solve counting problems using combinations.

The Basics of Counting Section 6.1.

Main Menu Main Menu (Click on the topics below) Permutation Example 1 Example 2 Circular Permutation Permuting r of n objects Example 1 Example 2 Example.

CSNB143 – Discrete Structure

Aaron Tan October Counting and Probability 1 IntroductionPossibility Trees and Multiplication RuleCounting Elements of Disjoint SetsThe Pigeonhole.

Discrete Mathematics Lecture # 25 Permutation & Combination.

Counting Principles Multiplication rule Permutations Combinations.

Counting Techniques Tree Diagram Multiplication Rule Permutations Combinations.

Spring 2016 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University.

Chapter 7 – Counting Techniques CSNB 143 Discrete Mathematical Structures.

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.

Section Basic Counting Principles: The Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n 1.

CS 104: Discrete Mathematics

Week 9 - Friday.  What did we talk about last time?  Partial orders  Total orders  Basic probability  Event  Sample space  Monty Hall  Multiplication.

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

Section 6.3. Section Summary Permutations Combinations.

The Multiplication Rule

Permutations and Combinations

Discrete Mathematics Lecture 8 Probability and Counting

4-1 Chapter 4 Counting Techniques.

CSNB 143 Discrete Mathematical Structures

Generalized Permutations and Combinations

Lesson 13.2 Find Probabilities Using Permutations

9. Counting and Probability 1

combinaTorial Analysis

Chapter 0.4 Counting Techniques.

4-1 Chapter 4 Counting Techniques.

4-1 Chapter 4 Counting Techniques.

Elementary Statistics

CS100: Discrete structures

Lesson 11.6 – 11.7 Permutations and Combinations

COUNTING AND PROBABILITY

Warm Up Permutations and Combinations Evaluate  4  3  2  1

Section 12.8 The Counting Principle and Permutations

COUNTING AND PROBABILITY

Lesson 11-1 Permutations and Combinations

Permutations and Combinations

COUNTING AND PROBABILITY

Permutations and Combinations

4-1 Chapter 4 Counting Techniques.

Bellwork Practice Packet 10.3 B side #3.

Permutations and Combinations

Standard DA-5.2 Objective: Apply permutations and combinations to find the number of possibilities of an outcome.

Permutations and Combinations

Lecture 7: Permutations and Combinations

Presentation transcript:

Discrete Structures Counting

Number of elements in a list

Possibility trees & multiplication rule Possibilities for Tournament Play Teams A and B are to play each other repeatedly until one wins two games in a row or a total of three games. One way in which this tournament can be played is for A to win the first game, B to win the second, and A to win the third and fourth games. Denote this by writing A–B–A–A. How many ways can the tournament be played?

Multiplication rule

Example A typical PIN (personal identification number) is a sequence of any four symbols chosen from the 26 letters in the alphabet and the ten digits, with repetition allowed. How many different PINs are possible? Now suppose that repetition is not allowed. a. How many different PINs are there? How many bit strings of length seven are there?

When the Multiplication Rule Is Difficult or Impossible to Apply Consider the following problem: Three officers—a president, a treasurer, and a secretary—are to be chosen from among four people: Ann, Bob, Cyd, and Dan. Suppose that, for various reasons, Ann cannot be president and either Cyd or Dan must be secretary.How many ways can the officers be chosen?

Permutations A permutation of a set of objects is an ordering of the objects in a row. For example, the set of elements a, b, and c has six permutations. abc acb cba bac bca cab In general, given a set of n objects, how many permutations does the set have? Imagine forming a permutation as an n-step operation: Step 1: Choose an element to write first. Step 2: Choose an element to write second. ... Step n: Choose an element to write nth.

Examples a. How many ways can the letters in the word COMPUTER be arranged in a row? b. How many ways can the letters in the word COMPUTER be arranged if the letters CO must remain next to each other (in order) as a unit? c. If letters of the word COMPUTER are randomly arranged in a row, what is the probability that the letters CO remain next to each other (in order) as a unit?

Permutations of Selected Element

Example a. How many different ways can three of the letters of the word BYTES be chosen and written in a row? b. How many different ways can this be done if the first letter must be B?

Addition Rule

Examples Suppose variable names in a programming language can be either a single uppercase letter or an uppercase letter followed by a digit. Find the number of possible variable names. Each user on a computer system has a password which must be six to eight characters long. Each character is an uppercase letter or digit. Each password must contain at least one digit. How many possible passwords are there?

Examples How many bit strings of length 8 either start with a 1 bit or end with the two bits 00?