Permutations and Combinations
|
|
|
- Steven Bradford
- 5 years ago
- Views:
Transcription
1 Permutations and Combinations Rosen, Chapter 5.3 Motivating question In a family of 3, how many ways can we arrange the members of the family in a line for a photograph? 1
2 Permutations A permutation of a set of distinct objects is an ordered arrangement of these objects. Example: (1, 3, 2, 4) is a permutation of the numbers 1, 2, 3, 4 How many permutations of n objects are there? How many permutations? How many permutations of n objects are there? Using the product rule: n. (n 1). (n 2),, 2. 1 = n! 2
3 Anagrams Anagram: a word, phrase, or name formed by rearranging the letters of another. Examples: cinema is an anagram of iceman "Tom Marvolo Riddle" = "I am Lord Voldemort The anagram server: Example How many ways can we arrange 4 students in a line for a picture? 3
4 Example How many ways can we arrange 4 students in a line for a picture? 4 possibilities for the first position, 3 for the second, 2 for the third, 1 for the fourth. 4*3*2*1 = 24 Example How many ways can we select 3 students from a group of 5 students to stand in line for a picture? 4
5 Example How many ways can we select 3 students from a group of 5 students to stand in line for a picture? 5 possibilities for the first person, 4 possibilities for the second, 3 for the third. 5*4*3 = 60 Definitions permutation a permutation of a set of distinct objects is an ordered arrangement of these objects. r-permutation a ordered arrangement of r elements of a set of objects. 5
6 Iclicker Question #1 You invite 4 people for a dinner party. How many different ways can they arrive assuming they enter separately? A) 6 (3!) B) 24 (4!) C) 120 (5!) D) 16 (n 2 ) E) 32 (2n 2 ) Iclicker Question #1 Answer You invite 4 people for a dinner party. How many different ways can they arrive assuming they enter separately? A) 6 (3!) B) 24 (4!) C) 120 (5!) D) 16 (n 2 ) E) 32 (2n 2 ) 6
7 Example In how many ways can a photographer at a wedding arrange six people in a row, (including the bride and groom)? Example In how many ways can a photographer at a wedding arrange six people in a row, (including the bride and groom)? 6! = 720 7
8 IClicker Question #2 In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if the bride must be next to the groom? A. 6! B. 5! C. 2X5! D. 2X6! E. 6! 5! IClicker Question #2 In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if the bride must be next to the groom? A. 6! B. 5! C. 2X5! D. 2X6! E. 6! 5! 8
9 IClicker Question #2 Answer In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if the bride must be next to the groom? Why? The bride and groom become a single unit which can be ordered 2 ways. IClicker Question #3 In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if the bride is not next to the groom? A. 6! B. 2X5! C. 2X6! D. 6! 5! E. 6! 2*5! 9
10 IClicker Question #3 Answer In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if the bride is not next to the groom? A. 6! B. 2X5! C. 2X6! D. 6! 5! E. 6! 2*5! IClicker Question #3 Answer In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if the bride is not next to the groom? Why? 6! possible ways for 6 2*5! - possible ways the bride is next to the groom. 10
11 Example In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if the bride s mother is positioned somewhere to the left of the groom? Example In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if the bride s mother is positioned somewhere to the left of the groom? 5! + (4 4!) + (3 4!) + (2 4!) + (1 4!) = = 360 possible photo arrangements in which the bride s mother is to the left of the groom. 11
12 Example The first position to fill is the position of the groom. At each position, the bride s mother can only occupy 1 of the slots to the left, the other 4 can be arranged in any manner. 5! + (4 4!) + (3 4!) + (2 4!) + (1 4!) = = 360 possible photo arrangements in which the bride s mother is to the left of the groom. Example 12
13 Iclicker Question #4 Count the number of ways to arrange n men and n women in a line so that no two men are next to each other and no two women are next to each other. A. n! B. 2 * n! C. n! * n! D. 2 * n! * n! Iclicker Question Answer #4 Count the number of ways to arrange n men and n women in a line so that no two men are next to each other and no two women are next to each other. A. n! B. 2 * n! C. n! * n! D. 2 * n! * n! 13
14 Why? Count the number of ways to arrange n men and n women in a line so that no two men are next to each other and no two women are next to each other. n! ways of representing men (n!) n! ways of representing women (n! * n!) Can start with either a man or a woman (x2) So 2*n!*n! The Traveling Salesman Problem (TSP) TSP: Given a list of cities and their pairwise distances, find a shortest possible tour that visits each city exactly once. Objective: find a permutation a 1,,a n of the cities that minimizes where d(i, j) is the distance between cities i and j An optimal TSP tour through Germany s 15 largest cities 14
15 Solving TSP Go through all permutations of cities, and evaluate the sum-of-distances, keeping the optimal tour. Need a method for generating all permutations Note: how many solutions to a TSP problem with n cities? Generating Permutations Let's design a recursive algorithm for generating all permutations of {0,1,2,,n-1}. 15
16 Generating Permutations Let's design a recursive algorithm for generating all permutations of {0,1,2,,n-1}. Starting point: decide which element to put first what needs to be done next? what is the base case? Solving TSP Is our algorithm for TSP that considers all permutations of n-1 elements a feasible one for solving TSP problems with hundreds or thousands of cities? 16
17 r-permutations An ordered arrangement of r elements of a set: number of r-permutations of a set with n elements: P(n,r) Example: List the 2-permutations of {a,b,c}. (a,b), (a,c), (b,a), (b,c), (c,a), (c,b) P(3,2) = 3 x 2 = 6 The number of r-permutations of a set of n elements: then there are P(n,r) = n(n 1) (n r + 1) (0 r n) Can be expressed as: P(n, r) = n! / (n r)! Note that P(n, 0) = 1. Iclicker Question #5 How many ways are there to select a first prize winner, a second prize winner, and a third prize winner from 100 different people who have entered a contest. A. 100! / 97! B. 100! C. 97! D. 100! 97! E
18 Iclicker Question #5 Answer How many ways are there to select a first prize winner, a second prize winner, and a third prize winner from 100 different people who have entered a contest. A. 100! / 97! B. 100! C. 97! D. 100! 97! E Iclicker Question #1 How many permutations of the letters ABCDEFGH contain the string ABC A. 6! B. 7! C. 8! D. 8!/5! E. 8!/6! 18
19 Iclicker Question #1 Answer How many permutations of the letters ABCDEFGH contain the string ABC A. 6! B. 7! C. 8! D. 8!/5! E. 8!/6! Iclicker Question #1 Answer How many permutations of the letters ABCDEFGH contain the string ABC Why? For the string ABC to appear, it can be treated as a single entity. That means there are 6 entities ABC, D, E, F, G, H. 19
20 Iclicker Question #2 Suppose there are 8 runners in a race. The winner receives a gold medal, the 2 nd place finisher a silver medal, 3 rd place a bronze, 4 th place a wooden medal. How many possible ways are there to award these medals? A. 5! B. 7! C. 8! / 4! D. 8! / 5! E. 8! / 6! Iclicker Question #2 Answer Suppose there are 8 runners in a race. The winner receives a gold medal, the 2 nd place finisher a silver medal, 3 rd place a bronze, 4 th place a wooden medal. How many possible ways are there to award these medals? A. 5! B. 7! C. 8! / 4! P(8,4) = 8!/(8-4)! = 8 * 7 * 6 * 5 D. 8! / 5! E. 8! / 6! 20
21 Combinations How many poker hands (five cards) can be dealt from a deck of 52 cards? How is this different than r-permutations? In an r-permutation we cared about order. In this case we don t Combinations An r-combination of a set is a subset of size r The number of r-combinations out of a set with n elements is denoted as C(n,r) or {1,3,4} is a 3-combination of {1,2,3,4} How many 2-combinations of {a,b,c,d}? 21
22 r-combinations How many r-combinations? Note that C(n, 0) = 1 C(n,r) satisfies: We can see that easily without using the formula Unordered versus ordered selections Two ordered selections are the same if the elements chosen are the same; the elements chosen are in the same order. Ordered selections: r-permutations. Two unordered selections are the same if the elements chosen are the same. (regardless of the order in which the elements are chosen) Unordered selections: r-combinations
23 Relationship between P(n,r) and C(n,r) Suppose we want to compute P(n,r). Constructing an r-permutation from a set of n elements can be thought as a 2-step process: Step 1: Choose a subset of r elements; Step 2: Choose an ordering of the r-element subset. Step 1 can be done in C(n,r) different ways. Step 2 can be done in r! different ways. Based on the multiplication rule, P(n,r) = C(n,r) r! Thus C( n, r) P( n, r) r! n! r! ( n r)! 45 Iclicker Question #3 How many poker hands (five cards) can be dealt from a deck of 52 cards? A. 52! B. 52! / 47! C. (52! 5!) / 47! D. 52! / (5! * 47!) E. (52! 47!) / 5! 23
24 Iclicker Question #3 Answer How many poker hands (five cards) can be dealt from a deck of 52 cards? A. 52! B. 52! / 47! C. (52! 5!) / 47! D. 52! / (5! * 47!) E. (52! 47!) / 5! Why? There are 52! / 47! Permutations so P(52,5) = 52! 47! Since order doesn t matter, there are 5! solutions that are considered identical. C(52,5) = 52! 5! * 47! 24
25 Iclicker Question #4 How many committees of 3 students can be formed from a group of 5 students? A. 5! / 2! B. 5! / (2! * 1!) C. 5! / (3! * 2!) D. 6! Iclicker Question #4 Answer How many committees of 3 students can be formed from a group of 5 students? A. 5! / 2! B. 5! / (2! * 1!) C. 5! / (3! * 2!) D. 6! 25
26 Iclicker Question #5 The faculty in biology and computer science want to develop a program in computational biology. A committee of 4 composed of two biologists and two computer scientists is tasked with doing this. How many such committees can be assembled out of 20 CS faculty and 30 biology faculty? A. C(50,4) B. C(600,4) C. C(20,2) + C(30,2) D. C(20,2) / C(30,2) E. C(20,2) * C(30,2) Iclicker Question #5 Answer The faculty in biology and computer science want to develop a program in computational biology. A committee of 4 composed of two biologists and two computer scientists is tasked with doing this. How many such committees can be assembled out of 20 CS faculty and 30 biology faculty? A. C(50,4) B. C(600,4) C. C(20,2) + C(30,2) D. C(20,2) / C(30,2) E. C(20,2) * C(30,2) 26
27 Why? There are C(20,2) combinations of CS There are C(30,2) combinations of Biology Using the product rule the total combinations is: C(20,2) * C(30,2) IClicker Question #6 A coin is flipped 10 times, producing either heads or tails. How many possible outcomes are there in total? A. 20 B C. 10 D /4 E. 2 27
28 IClicker Question #6 Answer A coin is flipped 10 times, producing either heads or tails. How many possible outcomes are there in total? A. 20 B C. 10 D /4 E. 2 IClicker Question #7 A coin is flipped 10 times, producing either heads or tails. How many possible outcomes contain exactly two heads? A. 2 8 B. 2 2 C /(2 8 X 2 2 ) D. 10!/(2! X 8!) E. 10!/2! 28
29 IClicker Question #7 Answer A coin is flipped 10 times, producing either heads or tails. How many possible outcomes contain exactly two heads? A. 2 8 B. 2 2 C /(2 8 X 2 2 ) D. 10!/(2! X 8!) E. 10!/2! IClicker Question #8 A coin is flipped 10 times, producing either heads or tails. How many possible outcomes contain at most 3 tails? A. 45 B. 175 C. 176 D. 251 E
30 IClicker Question #8 Answer A coin is flipped 10 times, producing either heads or tails. How many possible outcomes contain at most 3 tails? A. 45 B. 175 C. 176 D. 251 E. 252 IClicker Question #8 Answer A coin is flipped 10 times, producing either heads or tails. How many possible outcomes contain at most 3 tails? 1 way for 0 tails 10 ways for 1 tail 10!/2!*8! = 90/2 = 45 ways for 2 tails 10!/3!*7! = 720/6 = 120 ways for 3 tails 30
31 IClicker Question #9 A coin is flipped 10 times, producing either heads or tails. How many possible outcomes contain the same number of heads and tails? A / 2 B. 10! / (5! X 5!) C. 10! / 5! D. 10 / 5 IClicker Question #9 Answer A coin is flipped 10 times, producing either heads or tails. How many possible outcomes contain the same number of heads and tails? A / 2 B. 10! / (5! X 5!) C. 10! / 5! D. 10 / 5 31
32 Spock s Dilemma While commanding the Enterprise, Spock has 10 possible planets. He needs you to write a program that allows him to know how many possible combinations there are if he can only visit 4 of the planets. He has become fascinated with recursion and wants you to solve it recursively. How can we do this? Computing C(n, k) recursively consider the nth object C(n,k) = C(n-1,k-1) + C(n-1,k) pick n or don't 32
33 C(n, k): base case C(k, k) = 1 Why? C(n, 0) = 1 Why? Computing C(n, k) recursively C(n,k) = C(n-1,k-1) + C(n-1,k) pick n or don't C(k,k) = 1 C(n,0) = 1 we can easily code this as a recursive method! This is an example of a recurrence relation, which is a recursive mathematical expression 33
34 Some Advice about Counting Apply the multiplication rule if The elements to be counted can be obtained through a multistep selection process. Each step is performed in a fixed number of ways regardless of how preceding steps were performed. Apply the addition rule if The set of elements to be counted can be broken up into disjoint subsets Apply the inclusion/exclusion rule if It is simple to over-count and then to subtract duplicates 67 Some more advice about Counting Make sure that 1) every element is counted; 2) no element is counted more than once. (avoid double counting) When using the addition rule: 1) every outcome should be in some subset; 2) the subsets should be disjoint; if they are not, subtract the overlaps 68 34
Permutations and Combinations
Motivating question Permutations and Combinations A) Rosen, Chapter 5.3 B) C) D) Permutations A permutation of a set of distinct objects is an ordered arrangement of these objects. : (1, 3, 2, 4) is a
Discrete Structures Lecture Permutations and Combinations
Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these
The Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n ways to do the first task and n
Chapter 5 Chapter Summary 5.1 The Basics of Counting 5.2 The Pigeonhole Principle 5.3 Permutations and Combinations 5.5 Generalized Permutations and Combinations Section 5.1 The Product Rule The Product
Section Summary. Permutations Combinations Combinatorial Proofs
Section 6.3 Section Summary Permutations Combinations Combinatorial Proofs Permutations Definition: A permutation of a set of distinct objects is an ordered arrangement of these objects. An ordered arrangement
With Question/Answer Animations. Chapter 6
With Question/Answer Animations Chapter 6 Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations Binomial Coefficients and Identities Generalized Permutations and
Sec$on Summary. Permutations Combinations Combinatorial Proofs
Section 6.3 Sec$on Summary Permutations Combinations Combinatorial Proofs 2 Coun$ng ordered arrangements Ex: How many ways can we select 3 students from a group of 5 students to stand in line for a picture?
Board Question 1. There are 5 Competitors in 100m final. How many ways can gold silver and bronze be awarded? May 27, / 28
Board Question 1 There are 5 Competitors in 100m final. How many ways can gold silver and bronze be awarded? Photograph of Usain Bolt running a race removed due to copyright restrictions. May 27, 2014
Exercises Exercises. 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}?
Exercises Exercises 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}? 3. How many permutations of {a, b, c, d, e, f, g} end with
Discrete Mathematics: Logic. Discrete Mathematics: Lecture 15: Counting
Discrete Mathematics: Logic Discrete Mathematics: Lecture 15: Counting counting combinatorics: the study of the number of ways to put things together into various combinations basic counting principles
Counting. Chapter 6. With Question/Answer Animations
. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Counting Chapter
CPCS 222 Discrete Structures I Counting
King ABDUL AZIZ University Faculty Of Computing and Information Technology CPCS 222 Discrete Structures I Counting Dr. Eng. Farag Elnagahy farahelnagahy@hotmail.com Office Phone: 67967 The Basics of counting
Topics to be covered
Basic Counting 1 Topics to be covered Sum rule, product rule, generalized product rule Permutations, combinations Binomial coefficients, combinatorial proof Inclusion-exclusion principle Pigeon Hole Principle
Foundations of Computing Discrete Mathematics Solutions to exercises for week 12
Foundations of Computing Discrete Mathematics Solutions to exercises for week 12 Agata Murawska (agmu@itu.dk) November 13, 2013 Exercise (6.1.2). A multiple-choice test contains 10 questions. There are
Math 166: Topics in Contemporary Mathematics II
Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University September 30, 2017 Xin Ma (TAMU) Math 166 September 30, 2017 1 / 11 Last Time Factorials For any natural number n, we define
Sec.on Summary. The Product Rule The Sum Rule The Subtraction Rule (Principle of Inclusion- Exclusion)
Chapter 6 1 Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations Binomial Coefficients and Identities Generalized Permutations and Combinations 2 Section 6.1 3
November 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
CSCI FOUNDATIONS OF COMPUTER SCIENCE
1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 April 2, 2015 2 Announcements Homework 6 is due next Monday, April 6 at 10am in class. Homework 6 ClarificaMon In Problem 2C, where you need to
Combinatorial Proofs
Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A
COUNTING AND PROBABILITY
CHAPTER 9 COUNTING AND PROBABILITY Copyright Cengage Learning. All rights reserved. SECTION 9.2 Possibility Trees and the Multiplication Rule Copyright Cengage Learning. All rights reserved. Possibility
November 8, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 8, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Crystallographic notation The first symbol
CS100: DISCRETE STRUCTURES. Lecture 8 Counting - CH6
CS100: DISCRETE STRUCTURES Lecture 8 Counting - CH6 Lecture Overview 2 6.1 The Basics of Counting: THE PRODUCT RULE THE SUM RULE THE SUBTRACTION RULE THE DIVISION RULE 6.2 The Pigeonhole Principle. 6.3
Theory of Probability - Brett Bernstein
Theory of Probability - Brett Bernstein Lecture 3 Finishing Basic Probability Review Exercises 1. Model flipping two fair coins using a sample space and a probability measure. Compute the probability of
CS1800: Permutations & Combinations. Professor Kevin Gold
CS1800: Permutations & Combinations Professor Kevin Gold Permutations A permutation is a reordering of something. In the context of counting, we re interested in the number of ways to rearrange some items.
If a regular six-sided die is rolled, the possible outcomes can be listed as {1, 2, 3, 4, 5, 6} there are 6 outcomes.
Section 11.1: The Counting Principle 1. Combinatorics is the study of counting the different outcomes of some task. For example If a coin is flipped, the side facing upward will be a head or a tail the
EECS 203 Spring 2016 Lecture 15 Page 1 of 6
EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including
Chapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
Sec 5.1 The Basics of Counting
1 Sec 5.1 The Basics of Counting Combinatorics, the study of arrangements of objects, is an important part of discrete mathematics. In this chapter, we will learn basic techniques of counting which has
Combinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Lecture Notes Counting 101 Note to improve the readability of these lecture notes, we will assume that multiplication takes precedence over division, i.e. A / B*C
Counting and Probability Math 2320
Counting and Probability Math 2320 For a finite set A, the number of elements of A is denoted by A. We have two important rules for counting. 1. Union rule: Let A and B be two finite sets. Then A B = A
Introductory Probability
Introductory Probability Combinations Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Assigning Objects to Identical Positions Denitions Committee Card Hands Coin Toss Counts
Course Learning Outcomes for Unit V
UNIT V STUDY GUIDE Counting Reading Assignment See information below. Key Terms 1. Combination 2. Fundamental counting principle 3. Listing 4. Permutation 5. Tree diagrams Course Learning Outcomes for
Jong C. Park Computer Science Division, KAIST
Jong C. Park Computer Science Division, KAIST Today s Topics Basic Principles Permutations and Combinations Algorithms for Generating Permutations Generalized Permutations and Combinations Binomial Coefficients
With Question/Answer Animations. Chapter 6
With Question/Answer Animations Chapter 6 Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations Binomial Coefficients and Identities Generalized Permutations and
Section : Combinations and Permutations
Section 11.1-11.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words
W = {Carrie (U)nderwood, Kelly (C)larkson, Chris (D)aughtry, Fantasia (B)arrino, and Clay (A)iken}
UNIT V STUDY GUIDE Counting Course Learning Outcomes for Unit V Upon completion of this unit, students should be able to: 1. Apply mathematical principles used in real-world situations. 1.1 Draw tree diagrams
Learning Objectives for Section 7.4 Permutations and Combinations. 7.4 Permutations and Combinations
Learning Objectives for Section 7.4 Permutations and Combinations The student will be able to set up and compute factorials. The student will be able to apply and calculate permutations. The student will
Mat 344F challenge set #2 Solutions
Mat 344F challenge set #2 Solutions. Put two balls into box, one ball into box 2 and three balls into box 3. The remaining 4 balls can now be distributed in any way among the three remaining boxes. This
November 11, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.
Chapter Permutations and Combinations. Section 4 Permutations and Combinations. Example. Definition of n Factorial (n!)
Chapter 7 Logic, Sets, and Counting Section 4 Permutations and Combinations 7.4 Permutations and Combinations For more complicated problems, we will need to develop two important concepts: permutations
MATH 2420 Discrete Mathematics Lecture notes
MATH 2420 Discrete Mathematics Lecture notes Series and Sequences Objectives: Introduction. Find the explicit formula for a sequence. 2. Be able to do calculations involving factorial, summation and product
Finite Math - Fall 2016
Finite Math - Fall 206 Lecture Notes - /28/206 Section 7.4 - Permutations and Combinations There are often situations in which we have to multiply many consecutive numbers together, for example, in examples
A Probability Work Sheet
A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair six-sided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we
Strings. A string is a list of symbols in a particular order.
Ihor Stasyuk Strings A string is a list of symbols in a particular order. Strings A string is a list of symbols in a particular order. Examples: 1 3 0 4 1-12 is a string of integers. X Q R A X P T is a
The next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such
Block 1 - Sets and Basic Combinatorics. Main Topics in Block 1:
Block 1 - Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.
Counting (Enumerative Combinatorics) X. Zhang, Fordham Univ.
Counting (Enumerative Combinatorics) X. Zhang, Fordham Univ. 1 Chance of winning?! What s the chances of winning New York Megamillion Jackpot!! just pick 5 numbers from 1 to 56, plus a mega ball number
Solutions for Exam I, Math 10120, Fall 2016
Solutions for Exam I, Math 10120, Fall 2016 1. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 2, 3} B = {2, 4, 6, 8, 10}. C = {4, 5, 6, 7, 8}. Which of the following sets is equal to (A B) C? {1, 2, 3,
Using a table: regular fine micro. red. green. The number of pens possible is the number of cells in the table: 3 2.
Counting Methods: Example: A pen has tip options of regular tip, fine tip, or micro tip, and it has ink color options of red ink or green ink. How many different pens are possible? Using a table: regular
Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1
Probability and Counting Techniques
Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each
CISC 1400 Discrete Structures
CISC 1400 Discrete Structures Chapter 6 Counting CISC1400 Yanjun Li 1 1 New York Lottery New York Mega-million Jackpot Pick 5 numbers from 1 56, plus a mega ball number from 1 46, you could win biggest
Permutations and Combinations. Quantitative Aptitude & Business Statistics
Permutations and Combinations Statistics The Fundamental Principle of If there are Multiplication n 1 ways of doing one operation, n 2 ways of doing a second operation, n 3 ways of doing a third operation,
Independent Events. If we were to flip a coin, each time we flip that coin the chance of it landing on heads or tails will always remain the same.
Independent Events Independent events are events that you can do repeated trials and each trial doesn t have an effect on the outcome of the next trial. If we were to flip a coin, each time we flip that
MATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices
3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is
CHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events
CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes
Well, there are 6 possible pairs: AB, AC, AD, BC, BD, and CD. This is the binomial coefficient s job. The answer we want is abbreviated ( 4
2 More Counting 21 Unordered Sets In counting sequences, the ordering of the digits or letters mattered Another common situation is where the order does not matter, for example, if we want to choose a
Elementary Combinatorics
184 DISCRETE MATHEMATICAL STRUCTURES 7 Elementary Combinatorics 7.1 INTRODUCTION Combinatorics deals with counting and enumeration of specified objects, patterns or designs. Techniques of counting are
Discrete mathematics
Discrete mathematics Petr Kovář petr.kovar@vsb.cz VŠB Technical University of Ostrava DiM 470-2301/02, Winter term 2018/2019 About this file This file is meant to be a guideline for the lecturer. Many
PS 3.8 Probability Concepts Permutations & Combinations
BIG PICTURE of this UNIT: How can we visualize events and outcomes when considering probability events? How can we count outcomes in probability events? How can we calculate probabilities, given different
Permutations and Combinations Section
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Permutations and Combinations Section 13.3-13.4 Dr. John Ehrke Department of Mathematics Fall 2012 Permutations A permutation
Permutations. 1) How many unique 3 digit codes can be created from the 5 digits {1, 2, 3, 4, 5} if repeats are possible?
Name: WORKSHEET : Date: Permutations 1) How many unique 3 digit codes can be created from the 5 digits {1, 2, 3, 4, 5} if repeats are possible? 2) In how many ways can I create 2 letter words from the
{ a, b }, { a, c }, { b, c }
12 d.) 0(5.5) c.) 0(5,0) h.) 0(7,1) a.) 0(6,3) 3.) Simplify the following combinations. PROBLEMS: C(n,k)= the number of combinations of n distinct objects taken k at a time is COMBINATION RULE It can easily
Finite Math Section 6_4 Solutions and Hints
Finite Math Section 6_4 Solutions and Hints by Brent M. Dingle for the book: Finite Mathematics, 7 th Edition by S. T. Tan. DO NOT PRINT THIS OUT AND TURN IT IN!!!!!!!! This is designed to assist you in
Elementary Combinatorics CE 311S
CE 311S INTRODUCTION How can we actually calculate probabilities? Let s assume that there all of the outcomes in the sample space S are equally likely. If A is the number of outcomes included in the event
9.5 Counting Subsets of a Set: Combinations. Answers for Test Yourself
9.5 Counting Subsets of a Set: Combinations 565 H 35. H 36. whose elements when added up give the same sum. (Thanks to Jonathan Goldstine for this problem. 34. Let S be a set of ten integers chosen from
13.3 Permutations and Combinations
13.3 Permutations and Combinations There are 6 people who want to use an elevator. There is only room for 4 people. How many ways can 6 people try to fill this elevator (one at a time)? There are 6 people
CHAPTER 5 BASIC CONCEPTS OF PERMUTATIONS AND COMBINATIONS
CHAPTER 5 BASIC CONCEPTS OF PERMUTATIONS AND COMBINATIONS BASIC CONCEPTS OF PERM UTATIONS AND COM BINATIONS LEARNING OBJECTIVES After reading this Chapter a student will be able to understand difference
Counting and Probability
Counting and Probability What s to come? Probability. A bag contains: What is the chance that a ball taken from the bag is blue? Count blue. Count total. Divide. Today: Counting! Later this week: Probability.
CS 237: Probability in Computing
CS 237: Probability in Computing Wayne Snyder Computer Science Department Boston University Lecture 5: o Independence reviewed; Bayes' Rule o Counting principles and combinatorics; o Counting considered
More Probability: Poker Hands and some issues in Counting
More Probability: Poker Hands and some issues in Counting Data From Thursday Everybody flipped a pair of coins and recorded how many times they got two heads, two tails, or one of each. We saw that the
CSE 21 Mathematics for Algorithm and System Analysis
CSE 21 Mathematics for Algorithm and System Analysis Unit 1: Basic Count and List Section 3: Set CSE21: Lecture 3 1 Reminder Piazza forum address: http://piazza.com/ucsd/summer2013/cse21/hom e Notes on
MA 524 Midterm Solutions October 16, 2018
MA 524 Midterm Solutions October 16, 2018 1. (a) Let a n be the number of ordered tuples (a, b, c, d) of integers satisfying 0 a < b c < d n. Find a closed formula for a n, as well as its ordinary generating
MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability. Preliminary Concepts, Formulas, and Terminology
MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability Preliminary Concepts, Formulas, and Terminology Meanings of Basic Arithmetic Operations in Mathematics Addition: Generally
Chapter 3: Elements of Chance: Probability Methods
Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 3-4 2014-2015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,
CS1800: More Counting. Professor Kevin Gold
CS1800: More Counting Professor Kevin Gold Today Dealing with illegal values Avoiding overcounting Balls-in-bins, or, allocating resources Review problems Dealing with Illegal Values Password systems often
Counting in Algorithms
Counting Counting in Algorithms How many comparisons are needed to sort n numbers? How many steps to compute the GCD of two numbers? How many steps to factor an integer? Counting in Games How many different
Probability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
17. Symmetries. Thus, the example above corresponds to the matrix: We shall now look at how permutations relate to trees.
7 Symmetries 7 Permutations A permutation of a set is a reordering of its elements Another way to look at it is as a function Φ that takes as its argument a set of natural numbers of the form {, 2,, n}
Section Introduction to Sets
Section 1.1 - Introduction to Sets Definition: A set is a well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase
Mathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015
1 Probability Axioms Let A,B,C be three arbitrary events. Find the probability of exactly one of these events occuring. Sample space S: {ABC, AB, AC, BC, A, B, C, }, and S = 8. P(A or B or C) = 3 8. note:
1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?
1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,
Compound Probability. Set Theory. Basic Definitions
Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic
3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,
JIGSAW ACTIVITY, TASK # Make sure your answer in written in the correct order. Highest powers of x should come first, down to the lowest powers.
JIGSAW ACTIVITY, TASK #1 Your job is to multiply and find all the terms in ( 1) Recall that this means ( + 1)( + 1)( + 1)( + 1) Start by multiplying: ( + 1)( + 1) x x x x. x. + 4 x x. Write your answer
STAT 430/510 Probability Lecture 1: Counting-1
STAT 430/510 Probability Lecture 1: Counting-1 Pengyuan (Penelope) Wang May 22, 2011 Introduction In the early days, probability was associated with games of chance, such as gambling. Probability is describing
Permutations and Combinations
Permutations and Combinations Introduction Permutations and combinations refer to number of ways of selecting a number of distinct objects from a set of distinct objects. Permutations are ordered selections;
Name: Exam 1. September 14, 2017
Department of Mathematics University of Notre Dame Math 10120 Finite Math Fall 2017 Name: Instructors: Basit & Migliore Exam 1 September 14, 2017 This exam is in two parts on 9 pages and contains 14 problems
LAMC Junior Circle January 22, Oleg Gleizer. The Hanoi Tower. Part 2
LAMC Junior Circle January 22, 2012 Oleg Gleizer The Hanoi Tower Part 2 Definition 1 An algorithm is a finite set of clear instructions to solve a problem. An algorithm is called optimal, if the solution
Generalized Permutations and The Multinomial Theorem
Generalized Permutations and The Multinomial Theorem 1 / 19 Overview The Binomial Theorem Generalized Permutations The Multinomial Theorem Circular and Ring Permutations 2 / 19 Outline The Binomial Theorem
CH 13. Probability and Data Analysis
11.1: Find Probabilities and Odds 11.2: Find Probabilities Using Permutations 11.3: Find Probabilities Using Combinations 11.4: Find Probabilities of Compound Events 11.5: Analyze Surveys and Samples 11.6:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More 9.-9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on
Counting Methods and Probability
CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You
Examples: Experiment Sample space
Intro to Probability: A cynical person once said, The only two sure things are death and taxes. This philosophy no doubt arose because so much in people s lives is affected by chance. From the time a person
Math 42, Discrete Mathematics
c Fall 2018 last updated 10/29/2018 at 18:22:13 For use by students in this class only; all rights reserved. Note: some prose & some tables are taken directly from Kenneth R. Rosen, and Its Applications,
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis
![UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis](/thumbs/87/96626175.jpg "UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis")
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis Lecture 3 Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/ Lecture 3 Notes Goal for today: CL Section 3 Subsets,
CS 237 Fall 2018, Homework SOLUTION
0//08 hw03.solution.lenka CS 37 Fall 08, Homework 03 -- SOLUTION Due date: PDF file due Thursday September 7th @ :59PM (0% off if up to 4 hours late) in GradeScope General Instructions Please complete
Section 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
CSE 312: Foundations of Computing II Quiz Section #2: Combinations, Counting Tricks (solutions)
CSE 312: Foundations of Computing II Quiz Section #2: Combinations, Counting Tricks (solutions Review: Main Theorems and Concepts Combinations (number of ways to choose k objects out of n distinct objects,
Lecture 14. What s to come? Probability. A bag contains:
Lecture 14 What s to come? Probability. A bag contains: What is the chance that a ball taken from the bag is blue? Count blue. Count total. Divide. Today: Counting! Later: Probability. Professor Walrand.