# A1. Basic Reviews PERMUTATIONS and COMBINATIONS... or HOW TO COUNT

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 A1. Basic Reviews Appendix / A1. Basic Reviews / Perms & Combos-1 PERMUTATIONS and COMBINATIONS... or HOW TO COUNT Question 1: Suppose we wish to arrange n 5 people {a, b, c, d, e}, standing side by side, for a portrait. How many such distinct portraits ( permutations ) are possible? Example: a b c d e Here, every different ordering counts as a distinct permutation. For instance, the ordering (a,b,c,d,e) is distinct from (c,e,a,d,b), etc. Solution: There are 5 possible choices for which person stands in the first position (either a, b, c, d, or e). For each of these five possibilities, there are 4 possible choices left for who is in the next position. For each of these four possibilities, there are 3 possible choices left for the next position, and so on. Therefore, there are distinct permutations. See Table 1. This number, (or equivalently, ), is denoted by the symbol and read 5 factorial, so we can write the answer succinctly as 120. In general, FACT 1: The number of distinct PERMUTATIONS of n objects is n factorial, denoted by FACT 1: The number of distinct PERMUTATIONS of n objects is "n factorial", denoted by n! n, n! or equivalently, n, or equivalently, n (n-1) (n-2)... n 2 (n 1. 1) (n 2) Examples: 6! (by previous calculation) 720 3! ! ! 1 0! 1, BY CONVENTION (It may not be obvious why, but there are good mathematical reasons for it.)

2 Appendix / A1. Basic Reviews / Perms & Combos-2 Question 2: Now suppose we start with the same n 5 people {a, b, c, d, e}, but we wish to make portraits of only k 3 of them at a time. How many such distinct portraits are possible? Example: a b c Again, as above, every different ordering counts as a distinct permutation. For instance, the ordering (a,b,c) is distinct from (c,a,b), etc. Solution: By using exactly the same reasoning as before, there are permutations. See Table 2 for the explicit list! Note that this is technically NOT considered a factorial (since we don't go all the way down to 1), but we can express it as a ratio of factorials: (2 1) (2 1) 2!. In general, FACT 2: The number of distinct PERMUTATIONS of n objects, taken k at a time, is given by the ratio n! (n k)! n (n 1) (n 2)... (n k + 1). Question 3: Finally suppose that instead of portraits ( permutations ), we wish to form committees ( combinations ) of k 3 people from the original n 5. How many such distinct committees are possible? Example: c b a Now, every different ordering does NOT count as a distinct combination. For instance, the committee {a,b,c} is the same as the committee {c,a,b}, etc.

3 Appendix / A1. Basic Reviews / Perms & Combos-3 Solution: This time the reasoning is a little subtler. From the previous calculation, we know that # of permutations of k 3 from n 5 is equal to 2! 60. But now, all the ordered permutations of any three people (and there are 3! 6 of them, by FACT 1), will collapse into one single unordered combination, e.g., {a, b, c}, as illustrated. So... # of combinations of k 3 from n 5 is equal to 2!, divided by 3!, i.e., See Table 3 for the explicit list! This number, 3! 2!, is given the compact notation 5, read 5 choose 3, and corresponds to the 5 number of ways of selecting 3 objects from 5 objects, regardless of their order. Hence 10. In general, FACT 3: The number of distinct COMBINATIONS of n objects, taken k at a time, is given by the ratio n! k! (n k)! n (n 1) (n 2)... (n k + 1) k!. This quantity is usually written as n, and read n choose k. k Examples: 5 3! 2! 10, just done. Note that this is also equal to 5 2 2! 3! ! 2! 6! 8 7 6! 2! 6! Note that this is equal to 8 6 8! 6! 2! ! 14! 15 14! 1! 14! 15. Note that this is equal to Why? 7 7 7! 7! 0! 1. (Recall that 0! 1.) Note that this is equal to 7 1. Why? 0 Observe that it is neither necessary nor advisable to compute the factorials of large numbers directly. For instance, 8! 40320, but by writing it instead as 8 7 6!, we can cancel 6!, leaving only 8 7 above. Likewise, 14! cancels out of 1, leaving only 15, so we avoid having to compute 1, etc.

4 Appendix / A1. Basic Reviews / Perms & Combos-4 n Remark: is sometimes called a combinatorial symbol or binomial coefficient (in k connection with a fundamental mathematical result called the Binomial Theorem; you may also recall the related Pascal s Triangle ). The previous examples also show that binomial coefficients n n 5 possess a useful symmetry, namely,. For example,, but this is clearly k n k 3! 2! the same as 5. In other words, the number of ways of choosing 3-person committees 2 2! 3! from 5 people is equal to the number of ways of choosing 2-person committees from 5 people. A quick way to see this without any calculating is through the insight that every choice of a 3- person committee from a collection of 5 people leaves behind a 2-person committee, so the total number of both types of committee must be equal (10). Exercise: List all the ways of choosing 2 objects from 5, say {a, b, c, d, e}, and check these claims explicitly. That is, match each pair with its complementary triple in the list of Table 3. A Simple Combinatorial Application Suppose you toss a coin n 5 times in a row. How many ways can you end up with k 3 heads? Solution: The answer can be obtained by calculating the number of ways of rearranging 3 objects among 5; it only remains to determine whether we need to use permutations or combinations. Suppose, for example, that the 3 heads occur in the first three tosses, say a, b, and c, as shown below. Clearly, rearranging these three letters in a different order would not result in a different outcome. Therefore, different orderings of the letters a, b, and c should not count as distinct permutations, and likewise for any other choice of three letters among {a, b, c, d, e}. Hence, there 5 are 10 ways of obtaining k 3 heads in n 5 independent successive tosses. Exercise: Let H denote heads, and T denote tails. Using these symbols, construct the explicit list of 10 combinations. (Suggestion: Arrange this list of H/T sequences in alphabetical order. You should see that in each case, the three H positions match up exactly with each ordered triple in the list of Table 3. Why?) a b c d e

5 Appendix / A1. Basic Reviews / Perms & Combos-5 Table 1 Permutations of {a, b, c, d, e} These are the 120 ways of arranging 5 objects, in such a way that all the different orders count as being distinct. a b c d e b a c d e c a b d e d a b c e e a b c d a b c e d b a c e d c a b e d d a b e c e a b d c a b d c e b a d c e c a d b e d a c b e e a c b d a b d e c b a d e c c a d e b d a c e b e a c d b a b e c d b a e c d c a e b d d a e b c e a d b c a b e d c b a e d c c a e d b d a e c b e a d c b a c b d e b c a d e c b a d e d b a c e e b a c d a c b e d b c a e d c b a e d d b a e c e b a d c a c d b e b c d a e c b d a e d b c a e e b c a d a c d e b b c d e a c b d e a d b c e a e b c d a a c e b d b c e a d c b e a d d b e a c e b d a c a c e d b b c e d a c b e d a d b e c a e b d c a a d b c e b d a c e c d a b e d c a b e e c a b d a d b e c b d a e c c d a e b d c a e b e c a d b a d c b e b d c a e c d b a e d c b a e e c b a d a d c e b b d c e a c d b e a d c b e a e c b d a a d e b c b d e a c c d e a d d c e a b e c d a b a d e c b b d e c a c d e d a d c e b a e c d b a a e b c d b e a c d c e a b d d e a b c e d a b c a e b d c b e a d c c e a d b d e a c b e d a c b a e c b d b e c a d c e b a d d e b a c e d b a c a e c d b b e c d a c e b d a d e b c a e d b c a a e d b c b e d a c c e d a b d e c a b e d c a b a e d c b b e d c a c e d b a d e c b a e d c b a

6 Appendix / A1. Basic Reviews / Perms & Combos-6 Table 2 Permutations of {a, b, c, d, e}, taken 3 at a time These are the 2! 60 ways of arranging 3 objects among 5, in such a way that different orders of any triple count as being distinct, e.g., the 3! 6 permutations of (a, b, c), shown below. a b c b a c c a b d a b e a b a b d b a d c a d d a c e a c a b e b a e c a e d a e e a d a c b b c a c b a d b a e b a a c d b c d c b d d b c e b c a c e b c e c b e d b e e b d a d b b d a c d a d c a e c a a d c b d c c d b d c b e c b a d e b d e c d e d c e e c d a e b b e a c e a d e a e d a a e c b e c c e b d e b e d b a e d b e d c e d d e c e d c Table 3 Combinations of {a, b, c, d, e}, taken 3 at a time If different orders of the same triple are not counted as being distinct, then their six permutations are lumped as one, e.g., {a, b, c}. Therefore, the total number of combinations is 1 6 of the original 60, or 10. Notationally, we express this as 1 3! of the original 2!, i.e., 3! 2!, or more neatly, as 5. 5 These 10 combinations are listed below. a b c a b d a b e a c d a c e a d e b c d b c e b d e c d e

### MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1,

THE BINOMIAL THEOREM Pascal s Triangle and the Binomial Expansion Consider the following binomial expansions: (p + q) 0 1, (p+q) 1 p+q, (p + q) p +pq + q, (p + q) 3 p 3 +3p q+3pq + q 3, (p + q) 4 p 4 +4p

### 3.1 Events, Sample Spaces, and Probability

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 6, 2012 Lecture 3: Probability 3.1 Events, Sample Spaces, and Probability Date of latest update: August 8

### 1 Introduction. 2 Basic Principles. 2.1 Multiplication Rule. [Ch 9] Counting Methods. 400 lecture note #9

400 lecture note #9 [Ch 9] Counting Methods 1 Introduction In many discrete problems, we are confronted with the problem of counting. Here we develop tools which help us counting. Examples: o [9.1.2 (p.

### Sets, Venn Diagrams & Counting

MT 142 College Mathematics Sets, Venn Diagrams & Counting Module SC Terri Miller revised January 5, 2011 What is a set? Sets set is a collection of objects. The objects in the set are called elements of

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 /

### 94 Counting Solutions for Chapter 3. Section 3.2

94 Counting 3.11 Solutions for Chapter 3 Section 3.2 1. Consider lists made from the letters T, H, E, O, R, Y, with repetition allowed. (a How many length-4 lists are there? Answer: 6 6 6 6 = 1296. (b

### An Interesting Way to Combine Numbers

An Interesting Way to Combine Numbers Joshua Zucker and Tom Davis November 28, 2007 Abstract This exercise can be used for middle school students and older. The original problem seems almost impossibly

### Row Echelon Form and Reduced Row Echelon Form

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event

An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the

### Patterns & Math in Pascal s Triangle!

Uses for Pascal s Triangle An entry in Pascal's Triangle is usually given a row number and a place in that row, beginning with row zero and place, or element, zero. For instance, at the top of Pascal's

### Discrete probability and the laws of chance

Chapter 8 Discrete probability and the laws of chance 8.1 Introduction In this chapter we lay the groundwork for calculations and rules governing simple discrete probabilities. These steps will be essential

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### List of Standard Counting Problems

List of Standard Counting Problems Advanced Problem Solving This list is stolen from Daniel Marcus Combinatorics a Problem Oriented Approach (MAA, 998). This is not a mathematical standard list, just a

### Sequences and Series

Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will

### Math 408, Actuarial Statistics I, Spring 2008. Solutions to combinatorial problems

, Spring 2008 Word counting problems 1. Find the number of possible character passwords under the following restrictions: Note there are 26 letters in the alphabet. a All characters must be lower case

### SECTION 10-5 Multiplication Principle, Permutations, and Combinations

10-5 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial

### It is not immediately obvious that this should even give an integer. Since 1 < 1 5

Math 163 - Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some

### Combinations and Permutations

Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination

### Montana Common Core Standard

Algebra I Grade Level: 9, 10, 11, 12 Length: 1 Year Period(s) Per Day: 1 Credit: 1 Credit Requirement Fulfilled: A must pass course Course Description This course covers the real number system, solving

### NOTES ON COUNTING KARL PETERSEN

NOTES ON COUNTING KARL PETERSEN It is important to be able to count exactly the number of elements in any finite set. We will see many applications of counting as we proceed (number of Enigma plugboard

### Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

### Section 6.4: Counting Subsets of a Set: Combinations

Section 6.4: Counting Subsets of a Set: Combinations In section 6.2, we learnt how to count the number of r-permutations from an n-element set (recall that an r-permutation is an ordered selection of r

### 4. Binomial Expansions

4. Binomial Expansions 4.. Pascal's Triangle The expansion of (a + x) 2 is (a + x) 2 = a 2 + 2ax + x 2 Hence, (a + x) 3 = (a + x)(a + x) 2 = (a + x)(a 2 + 2ax + x 2 ) = a 3 + ( + 2)a 2 x + (2 + )ax 2 +

### 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

### Discrete Mathematics Lecture 5. Harper Langston New York University

Discrete Mathematics Lecture 5 Harper Langston New York University Empty Set S = {x R, x 2 = -1} X = {1, 3}, Y = {2, 4}, C = X Y (X and Y are disjoint) Empty set has no elements Empty set is a subset of

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### Notes on counting finite sets

Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................

### Discrete Mathematics and its Applications Counting (2) Xiaocong ZHOU

Discrete Mathematics and its Applications Counting (2) Xiaocong ZHOU Department of Computer Science Sun Yat-sen University Feb. 2016 http://www.cs.sysu.edu.cn/ zxc isszxc@mail.sysu.edu.cn Xiaocong ZHOU

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

### Lecture 6: Probability. If S is a sample space with all outcomes equally likely, define the probability of event E,

Lecture 6: Probability Example Sample Space set of all possible outcomes of a random process Flipping 2 coins Event a subset of the sample space Getting exactly 1 tail Enumerate Sets If S is a sample space

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### COUNTING SUBSETS OF A SET: COMBINATIONS

COUNTING SUBSETS OF A SET: COMBINATIONS DEFINITION 1: Let n, r be nonnegative integers with r n. An r-combination of a set of n elements is a subset of r of the n elements. EXAMPLE 1: Let S {a, b, c, d}.

### SOLUTIONS, Homework 6

SOLUTIONS, Homework 6 1. Testing out the identities from Section 6.4. (a) Write out the numbers in the first nine rows of Pascal s triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

### Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

### 1.4 Factors and Prime Factorization

1.4 Factors and Prime Factorization Recall from Section 1.2 that the word factor refers to a number which divides into another number. For example, 3 and 6 are factors of 18 since 3 6 = 18. Note also that

### Elements of probability theory

2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

### Lecture 11: Probability models

Lecture 11: Probability models Probability is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model we need the following ingredients A sample

### 1 Algebra - Partial Fractions

1 Algebra - Partial Fractions Definition 1.1 A polynomial P (x) in x is a function that may be written in the form P (x) a n x n + a n 1 x n 1 +... + a 1 x + a 0. a n 0 and n is the degree of the polynomial,

### The Binomial Probability Distribution

The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

### 16.1. Sequences and Series. Introduction. Prerequisites. Learning Outcomes. Learning Style

Sequences and Series 16.1 Introduction In this block we develop the ground work for later blocks on infinite series and on power series. We begin with simple sequences of numbers and with finite series

### 0018 DATA ANALYSIS, PROBABILITY and STATISTICS

008 DATA ANALYSIS, PROBABILITY and STATISTICS A permutation tells us the number of ways we can combine a set where {a, b, c} is different from {c, b, a} and without repetition. r is the size of of the

### Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

### Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

### Module 6: Basic Counting

Module 6: Basic Counting Theme 1: Basic Counting Principle We start with two basic counting principles, namely, the sum rule and the multiplication rule. The Sum Rule: If there are n 1 different objects

### The concept of probability is fundamental in statistical analysis. Theory of probability underpins most of the methods used in statistics.

Elementary probability theory The concept of probability is fundamental in statistical analysis. Theory of probability underpins most of the methods used in statistics. 1.1 Experiments, outcomes and sample

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### CHAPTER 9: DISCRETE MATH

CHAPTER 9: DISCRETE MATH (Chapter 9: Discrete Math) 9.01 Calculus tends to deal more with continuous mathematics than discrete mathematics. What is the difference? Analogies may help the most. Discrete

### Math 421: Probability and Statistics I Note Set 2

Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the

### Sets and set operations

CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

### Axiom A.1. Lines, planes and space are sets of points. Space contains all points.

73 Appendix A.1 Basic Notions We take the terms point, line, plane, and space as undefined. We also use the concept of a set and a subset, belongs to or is an element of a set. In a formal axiomatic approach

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Full and Complete Binary Trees

Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

### Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

### Equivalence relations

Equivalence relations A motivating example for equivalence relations is the problem of constructing the rational numbers. A rational number is the same thing as a fraction a/b, a, b Z and b 0, and hence

### Montana Common Core Standard

Algebra 2 Grade Level: 10(with Recommendation), 11, 12 Length: 1 Year Period(s) Per Day: 1 Credit: 1 Credit Requirement Fulfilled: Mathematics Course Description This course covers the main theories in

### 3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

### RELATIONS AND FUNCTIONS

Chapter 1 RELATIONS AND FUNCTIONS There is no permanent place in the world for ugly mathematics.... It may be very hard to define mathematical beauty but that is just as true of beauty of any kind, we

### Synthetic Projective Treatment of Cevian Nests and Graves Triangles

Synthetic Projective Treatment of Cevian Nests and Graves Triangles Igor Minevich 1 Introduction Several proofs of the cevian nest theorem (given below) are known, including one using ratios along sides

### Interpolating Polynomials Handout March 7, 2012

Interpolating Polynomials Handout March 7, 212 Again we work over our favorite field F (such as R, Q, C or F p ) We wish to find a polynomial y = f(x) passing through n specified data points (x 1,y 1 ),

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### The Inclusion-Exclusion Principle

Physics 116C Fall 2012 The Inclusion-Exclusion Principle 1. The probability that at least one of two events happens Consider a discrete sample space Ω. We define an event A to be any subset of Ω, which

### THE MULTINOMIAL DISTRIBUTION. Throwing Dice and the Multinomial Distribution

THE MULTINOMIAL DISTRIBUTION Discrete distribution -- The Outcomes Are Discrete. A generalization of the binomial distribution from only 2 outcomes to k outcomes. Typical Multinomial Outcomes: red A area1

### Chapter 5 - Probability

Chapter 5 - Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set

### Basic Probability Concepts

page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes

### 5. Conditional Expected Value

Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 5. Conditional Expected Value As usual, our starting point is a random experiment with probability measure P on a sample space Ω. Suppose that X is

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### Math 115 Spring 2011 Written Homework 5 Solutions

. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

### Unit 4 The Bernoulli and Binomial Distributions

PubHlth 540 4. Bernoulli and Binomial Page 1 of 19 Unit 4 The Bernoulli and Binomial Distributions Topic 1. Review What is a Discrete Probability Distribution... 2. Statistical Expectation.. 3. The Population

### Exam #5 Sequences and Series

College of the Redwoods Mathematics Department Math 30 College Algebra Exam #5 Sequences and Series David Arnold Don Hickethier Copyright c 000 Don-Hickethier@Eureka.redwoods.cc.ca.us Last Revision Date:

### Mathematical goals. Starting points. Materials required. Time needed

Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about

### CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we construct the set of rational numbers Q using equivalence

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected

### Solutions to Exercises 5

Discrete Mathematics Lent 009 MA0 Solutions to Exercises 5 () Let a n be the number of n-letter words formed from the 6 letters of the alphabet, in which the five vowels A, E, I, O, U together occur an

### Lecture 2 Binomial and Poisson Probability Distributions

Lecture 2 Binomial and Poisson Probability Distributions Binomial Probability Distribution l Consider a situation where there are only two possible outcomes (a Bernoulli trial) H Example: u flipping a

### MTH6109: Combinatorics. Professor R. A. Wilson

MTH6109: Combinatorics Professor R. A. Wilson Autumn Semester 2011 2 Lecture 1, 26/09/11 Combinatorics is the branch of mathematics that looks at combining objects according to certain rules (for example:

### MAT 1000. Mathematics in Today's World

MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities

### Discrete Mathematics. Stephen Perencevich

Discrete Mathematics Stephen Perencevich Stephen Perencevich Gonzaga College High School Washington, DC sperencevich@gonzaga.org c 0 All rights reserved. Algebra II: Discrete Mathematics Arithmetic Sequences

### JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability)

JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability) Author: Daniel Cooney Reviewer: Rachel Zax Last modified: January 4, 2012 Reading from Meyer, Mathematics

### 4.3. Addition and Multiplication Laws of Probability. Introduction. Prerequisites. Learning Outcomes. Learning Style

Addition and Multiplication Laws of Probability 4.3 Introduction When we require the probability of two events occurring simultaneously or the probability of one or the other or both of two events occurring

### Combinatorics. Chapter 1. 1.1 Factorials

Chapter 1 Combinatorics Copyright 2009 by David Morin, morin@physics.harvard.edu (Version 4, August 30, 2009) This file contains the first three chapters (plus some appendices) of a potential book on Probability

### Chapter 17: Aggregation

Chapter 17: Aggregation 17.1: Introduction This is a technical chapter in the sense that we need the results contained in it for future work. It contains very little new economics and perhaps contains

### 3. Examples of discrete probability spaces.. Here α ( j)

3. EXAMPLES OF DISCRETE PROBABILITY SPACES 13 3. Examples of discrete probability spaces Example 17. Toss n coins. We saw this before, but assumed that the coins are fair. Now we do not. The sample space

### Elementary probability

Elementary probability Many of the principal applications of calculus are to questions of probability and statistics. We shall include here an introduction to elementary probability, and eventually some

### WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM

WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM EXAMPLE 1. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the first head is observed.

### Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

### Online Materials: Counting Rules The Poisson Distribution

Chapter 6 The Poisson Distribution Counting Rules When the outcomes of a chance experiment are equally likely, one way to determine the probability of some event E is to calculate number of outcomes for

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### 5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0

a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Why is the number of 123- and 132-avoiding permutations equal to the number of binary trees?

Why is the number of - and -avoiding permutations equal to the number of binary trees? What are restricted permutations? We shall deal with permutations avoiding some specific patterns. For us, a permutation

### Sections 2.1, 2.2 and 2.4

SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains