Text: Brualdi, Introductory Combinatorics 5th Ed. Prof: Paul Terwilliger Selected solutions II for Chapter 2
|
|
- Denis Stephens
- 6 years ago
- Views:
Transcription
1 Math 7 Text: Brualdi, Introductory Combinatorics th Ed Prof: Paul Terwilliger Selected solutions II for Chapter 0 We proceed in stages: The answer is! pick gender to the parent s right order the girls clockwise! order the boys clockwise! Now assume that there are two parents, labelled P and Q Suppose we move clockwise around the table from P to Q Let n denote the number of seats between the two Thus n = 0 resp n = 0 if Q sits next to P at P s left resp right We now partition the set of solutions according to the value of n n # seatings 0!!!!!!! 7! 8! 9! 0! The answer is the sum of the entries in the right-most column, which comes to 0! 8 We make a change of variables Define y = x, y = x, y = x +, y = x 8 Note that {x i } is a solution to the original problem if and only if y 0, y 0, y 0, y 0, y + y + y + y = Therefore the number of solutions is + = 8 9 a There are 0 ways to choose six sticks from the twenty available sticks
2 b Label the sticks,,, 0 Suppose we choose six sticks labelled {x i } with x < x < < x Define y = x, y = x x, y = x x, y = x x, y = x x, y = x x, y 7 = 0 x Observe that the solutions {x i } to the original problem correspond to the integral solutions {y i } 7 for y i 0 i 7, 7 y i = 9 Therefore the number of solutions {x i } to the original problem is = 7 c We proceed as in b with the modification y = x, y = x x, y = x x, y = x x, y = x x, y = x x, y 7 = 0 x The solutions {x i } to the original problem correspond to the integral solutions {y i } 7 for y i 0 i 7, 7 y i = Therefore the number of solutions {x i } to the original problem is = 7 We proceed in stages: The answer is We proceed in stages: hand out the orange give one apple to each of the other children distribute remaining 0 apples to children hand out lemon drink hand out lime drink give one orange drink to each of the remaining two students distribute remaining 8 orange drinks to students
3 The answer is the product of the entries in the right-most column, which comes to a For i let x i denote the number of books on shelf i We seek the number of integral solutions to x i 0 i, x i = 0 The answer is 0 + = b We proceed in stages: put book on a shelf put book on a shelf 0 put book 0 on a shelf The answer is the product of the entries in the right-most column, which comes to 0 c We proceed in stages: order the books 0! pick a solution {x i } to part a put the first x books on shelf put the next x books on shelf put the next x books on shelf put the next x books on shelf 7 put the last x books on shelf The answer is the product of the entries in the right-most column, which comes to 0! a The word has 7 letters with repetitions letter A B D E H I K O P R S T mult
4 7!!! b The word has 9 letters with repetitions letter A C F H I L N O P T U mult 9 c The word has letters with repetitions 9!!!! 9! letter A C E I L M N O P R S T U V mult 9 d!!!!!! 9! 0 The number of ways to pick the bagels is equal to the number of integral solutions for x i 0 i, x i = which comes to + = 0 This is the denominator We now compute the numerator for the first probability The number of ways to pick the bagels so that you get at least one bagel of each kind is equal to the number of integral solutions for y i 0 i, y i = 9 which is 9+ = The first desired probability is 0
5 We now compute the numerator for the second probability The number of ways to pick the bagels so that you get at least three sesame bagels is equal to the number of integral solutions for z i 0 i, z i = which is + = 7 The second desired probability is 7 0 The sample space S satisfies The first event E satisfies S = 9! E = 9! 9 The first desired probability is E / S which comes to 9 The second event F satisfies F =!! The second desired probability is F / S which comes to!! 9! 9 The size of the sample space is This is the denominator for a e below a The die numbers must be some permutation of,,, ways or,,, ways The numerator is + = 0 The desired probability is 0/ b One dot occurs either once ways or twice ways or not at all ways The desired probability is + + c The desired probability is / d The desired probability is P, / e The die numbers must be some permutation of i, i, i, j ways or i, i, j, j ways Here we mean i j The desired probability is +
Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin,
Chapter 8 Exercises Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin, morin@physics.harvard.edu 8.1 Chapter 1 Section 1.2: Permutations 1. Assigning seats *
More informationBaye s theorem. Baye s Theorem Let E and F be two possible events of an experiment, then P (F ) P (E F ) P (F ) P (E F ) + P (F ) P (E F ).
Baye s Theorem Assume that you know the probability that a child will be born with blond hair given that both his parents have blond hair. You might also be interested in knowing the probability that a
More informationCarleton University. Final Examination Winter DURATION: 2 HOURS No. of students: 152
Carleton University Final Examination Winter 2014 DURATION: 2 HOURS No. of students: 152 Department Name & Course Number: Computer Science COMP 2804B Course Instructor: Michiel Smid Authorized memoranda:
More informationCounting. Math 301. November 24, Dr. Nahid Sultana
Basic Principles Dr. Nahid Sultana November 24, 2012 Basic Principles Basic Principles The Sum Rule The Product Rule Distinguishable Pascal s Triangle Binomial Theorem Basic Principles Combinatorics: The
More informationMath Review. Name:
Math 30-1 Name: Review 1. Given the graph of : Sketch the graph of the given transformation on the same grid Describe how the transformed graph relates to the graph of Write the equation of the image of
More informationInferential statistics
Inferential statistics Inference involves making a Generalization about a larger group of individuals on the basis of a subset or sample. Ahmed-Refat-ZU Null and alternative hypotheses In hypotheses testing,
More informationSECTION ONE - (3 points problems)
International Kangaroo Mathematics Contest 0 Student Level Student (Class & ) Time Allowed : hours SECTION ONE - ( points problems). The water level in a port city rises and falls on a certain day as shown
More informationMath Challengers Provincial 2016 Blitz and Bull s-eye rounds.
Math hallengers Provincial 016 litz and ull s-eye rounds. Solutions proposed by Sean Wang from Point Grey litz 1. What is the sum of all single digit primes? Solution: Recall that a prime number only has
More information6 A rectangular garden measures 38m by 20m, to the nearest metre. Calculate the maximum possible area of the garden.
Year Higher Homework Questions that require the use of a calculator are indicated by a Expand and Simplify ( x + )( x ) A bag contains red balls and blue balls. A ball is picked at random from the bag
More informationSenior Math Circles November 19, 2008 Probability II
University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles November 9, 2008 Probability II Probability Counting There are many situations where
More informationDescriptive Statistics Class Practice [133 marks]
Descriptive Statistics Class Practice [133 marks] The weekly wages (in dollars) of 80 employees are displayed in the cumulative frequency curve below. 1a. (i) (ii) Write down the median weekly wage. Find
More informationPrinciples of Counting. Debdeep Mukhopadhyay IIT Madras
Principles of Counting Debdeep Mukhopadhyay IIT Madras Part-I The Sum Rule Two tasks T 1 and T 2 are to be performed. If the task T 1 can be performed in m different ways and if the task T 2 can be performed
More informationMATH 10B METHODS OF MATHEMATICS: CALCULUS, STATISTICS AND COMBINATORICS
MATH 10B METHODS OF MATHEMATICS: CALCULUS, STATISTICS AND COMBINATORICS Lior Pachter and Lawrence C. Evans Department of Mathematics University of California Berkeley, CA 94720 January 21, 2013 Lior Pachter
More information9. DISCRETE PROBABILITY DISTRIBUTIONS
9. DISCRETE PROBABILITY DISTRIBUTIONS Random Variable: A quantity that takes on different values depending on chance. Eg: Next quarter s sales of Coca Cola. The proportion of Super Bowl viewers surveyed
More informationChapter 2 PROBABILITY SAMPLE SPACE
Chapter 2 PROBABILITY Key words: Sample space, sample point, tree diagram, events, complement, union and intersection of an event, mutually exclusive events; Counting techniques: multiplication rule, permutation,
More informationErrata for Introductory Combinatorics, 4th edition Author: Richard A. Brualdi
Errata for Introductory Combinatorics, 4th edition Author: Richard A Brualdi 1 Page ix, line 8 (Preface): Louis Deaett 2 Page 43, line 6-7 (Exercise 28): What is the smallest sum a 1 + a 2 + + a 100 that
More informationQ1. Amina is making designs with two different shapes.
Q1. Amina is making designs with two different shapes. She gives each shape a value. Total value is 147 Total value is 111 Calculate the value of each shape. Q2. n stands for a whole number. 2n is greater
More informationConditional Probability
Chapter 3 Conditional Probability 3.1 Definition of conditional probability In spite of our misgivings, let us persist with the frequency definition of probability. Consider an experiment conducted N times
More informationAn Introduction to Combinatorics
Chapter 1 An Introduction to Combinatorics What Is Combinatorics? Combinatorics is the study of how to count things Have you ever counted the number of games teams would play if each team played every
More information324 Stat Lecture Notes (1) Probability
324 Stat Lecture Notes 1 robability Chapter 2 of the book pg 35-71 1 Definitions: Sample Space: Is the set of all possible outcomes of a statistical experiment, which is denoted by the symbol S Notes:
More informationNNC Year 6 Algebra. 61 minutes. 59 marks. Page 1 of 32
NNC Year 6 Algebra 6 minutes 59 marks Page of 32 Q. Here is a sequence of towers built from cubes. These are the plans of each tower. The numbers show how many cubes are in each vertical column. How many
More informationAlgebra II Notes Unit Four: Matrices and Determinants
Syllabus Objectives: 4. The student will organize data using matrices. 4.2 The student will simplify matrix expressions using the properties of matrices. Matrix: a rectangular arrangement of numbers (called
More informationUC Berkeley, CS 174: Combinatorics and Discrete Probability (Fall 2008) Midterm 1. October 7, 2008
UC Berkeley, CS 74: Combinatorics and Discrete Probability (Fall 2008) Midterm Instructor: Prof. Yun S. Song October 7, 2008 Your Name : Student ID# : Read these instructions carefully:. This is a closed-book
More informationAlgebra I Notes Unit Thirteen: Rational Expressions and Equations
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Syllabus Objective: 10. The student will solve rational equations. (proportions and percents) Ratio: the relationship a b of two quantities,
More informationMATH 3C: MIDTERM 1 REVIEW. 1. Counting
MATH 3C: MIDTERM REVIEW JOE HUGHES. Counting. Imagine that a sports betting pool is run in the following way: there are 20 teams, 2 weeks, and each week you pick a team to win. However, you can t pick
More informationChapter 5 : Probability. Exercise Sheet. SHilal. 1 P a g e
1 P a g e experiment ( observing / measuring ) outcomes = results sample space = set of all outcomes events = subset of outcomes If we collect all outcomes we are forming a sample space If we collect some
More informationPrelim 1 (10 8)!8! = 10 9 = 45. (b) How many ways can it be done if the first three questions are required? 2 = 21
Math 1105 Fall 010 J.T. Gene Hwang, Instructor Robyn Miller, T.A. Prelim 1 Instructions This is a closed book exam. Graphing calculators are not permitted. Solutions should be written in your blue book
More informationSection A Number Facts
Grade Play! Mathematics Answer Book Section A Number Facts Concept -digit Numbers. Study: In 0 there are 0 units (ones). In there are units (ones). 0 + The value of the. The value of the. ten + units or
More information1. Applying the order of operations, we first compute 2 3 = 6, and thus = = 7.
1. Applying the order of operations, we first compute 2 3 = 6, and thus 1 + 2 3 = 1 + 6 = 7. 2. Since every flower has 7 petals, and Alex has 1001 petals, there are 1001 7 = 13 flowers. 3. Since 30 minutes
More informationEnd Of Term 2 Revision Sheet
Egyptian British International School Math Department Name:. Year (8..) End Of Term 2 Revision Sheet * Answer The following questions after revising your classwork copybook, course book, homework book
More informationGrade 6 Math Circles. Gauss Contest Preparation - Solutions
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles March 25/26, 2014 Gauss Contest Preparation - Solutions General Information The Gauss
More informationMATH section 4.4 Concavity and Curve Sketching Page 1. is increasing on I. is decreasing on I. = or. x c
MATH 0100 section 4.4 Concavity and Curve Sketching Page 1 Definition: The graph of a differentiable function y = (a) concave up on an open interval I if df f( x) (b) concave down on an open interval I
More informationRivervale Primary School Maths Workshop for P4 to P6 Parents. 10 February 2018, Saturday a.m.
Rivervale Primary School Maths Workshop for P4 to P6 Parents 10 February 2018, Saturday 10.15 11.45 a.m. MATHS ASSESSMENT, EXAM PAPER FORMAT AND APPROACH TO WORD PROBLEMS Assessments: Formative (Assessment
More informationSTUDY GUIDE Math 20. To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition
STUDY GUIDE Math 0 To the students: To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition When you study Algebra, the material is presented to you in a logical sequence.
More informationFinal Examination. Your name: Circle the name of your Tutorial Instructor: David Hanson Jelani Sayan
Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science December 21 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised December 22, 2005, 1118 minutes Circle
More informationCS 441 Discrete Mathematics for CS Lecture 19. Probabilities. CS 441 Discrete mathematics for CS. Probabilities
CS 441 Discrete Mathematics for CS Lecture 19 Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Experiment: a procedure that yields one of the possible outcomes Sample space: a set of possible outcomes
More informationExam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3)
1 Exam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3) On this exam, questions may come from any of the following topic areas: - Union and intersection of sets - Complement of
More information2. Counting and Probability
2. Counting and Probability 2.1.1 Factorials 2.1.2 Combinatorics 2.2.1 Probability Theory 2.2.2 Probability Examples 2.1.1 Factorials Combinatorics Combinatorics is the mathematics of counting. It can
More informationMath 378 Spring 2011 Assignment 4 Solutions
Math 3 Spring 2011 Assignment 4 Solutions Brualdi 6.2. The properties are P 1 : is divisible by 4. P 2 : is divisible by 6. P 3 : is divisible by. P 4 : is divisible by 10. Preparing to use inclusion-exclusion,
More informationMATH 3012 N Solutions to Review for Midterm 2
MATH 301 N Solutions to Review for Midterm March 7, 017 1. In how many ways can a n rectangular checkerboard be tiled using 1 and pieces? Find a closed formula. If t n is the number of ways to tile the
More informationSHOW ALL WORK FOR CREDIT!!!
MATH 1342ÞP04 FALL 2010 CHAPTERS 4, 5 & 6 NAME SHOW ALL WORK FOR CREDIT!!! 1. A study was done to investigate a possible relationship the circumference (in ft) and the height (in ft) of trees. The data
More informationWhat is the probability of getting a heads when flipping a coin
Chapter 2 Probability Probability theory is a branch of mathematics dealing with chance phenomena. The origins of the subject date back to the Italian mathematician Cardano about 1550, and French mathematicians
More informationData Presentation. Naureen Ghani. May 4, 2018
Data Presentation Naureen Ghani May 4, 2018 Data is only as good as how it is presented. How do you take hundreds or thousands of data points and create something a human can understand? This is a problem
More informationCombinatorial Analysis
Chapter 1 Combinatorial Analysis STAT 302, Department of Statistics, UBC 1 A starting example: coin tossing Consider the following random experiment: tossing a fair coin twice There are four possible outcomes,
More informationStatistical Theory 1
Statistical Theory 1 Set Theory and Probability Paolo Bautista September 12, 2017 Set Theory We start by defining terms in Set Theory which will be used in the following sections. Definition 1 A set is
More informationPOTENTIAL PROBLEM DESCRIPTIONS
POTENTIAL PROBLEM DESCRIPTIONS I. Combinatorics (a) Problem 1: Partitions We define a partition of a number, n, to be a sequence of non-increasing positive integers that sum to n. We want to examine the
More informationstudents all of the same gender. (Total 6 marks)
January Exam Review: Math 11 IB HL 1. A team of five students is to be chosen at random to take part in a debate. The team is to be chosen from a group of eight medical students and three law students.
More informationPrinciples of Mathematics 12
Principles of Mathematics 12 Examination Booklet 2007/08 Released Exam January 2008 Form B DO NOT OPEN ANY EXAMINATION MATERIALS UNTIL INSTRUCTED TO DO SO. FOR FURTHER INSTRUCTIONS REFER TO THE RESPONSE
More informationNMC Sample Problems: Grade 7
NMC Sample Problems: Grade 7. If Amy runs 4 4 mph miles in every 8 4. mph hour, what is her unit speed per hour? mph. mph 6 mph. At a stationary store in a state, a dozen of pencils originally sold for
More informationk-protected VERTICES IN BINARY SEARCH TREES
k-protected VERTICES IN BINARY SEARCH TREES MIKLÓS BÓNA Abstract. We show that for every k, the probability that a randomly selected vertex of a random binary search tree on n nodes is at distance k from
More informationP5/6 Mathematics Parents Workshop. 7 April 2018 Presented by : Mrs Janice Tan (ST) Mrs Karen Anne Silva Mr Nach
P5/6 Mathematics Parents Workshop 7 April 2018 Presented by : Mrs Janice Tan (ST) Mrs Karen Anne Silva Mr Nach What is Heuristics? Heuristics are general rules of thumb of what students can do to tackle
More informationChapter 6. Probability
Chapter 6 robability Suppose two six-sided die is rolled and they both land on sixes. Or a coin is flipped and it lands on heads. Or record the color of the next 20 cars to pass an intersection. These
More informationChapter 13, Probability from Applied Finite Mathematics by Rupinder Sekhon was developed by OpenStax College, licensed by Rice University, and is
Chapter 13, Probability from Applied Finite Mathematics by Rupinder Sekhon was developed by OpenStax College, licensed by Rice University, and is available on the Connexions website. It is used under a
More informationPrinciples of Mathematics 12
Principles of Mathematics 12 Examination Booklet Sample 2007/08 Form A DO NOT OPEN ANY EXAMINATION MATERIALS UNTIL INSTRUCTED TO DO SO. FOR FURTHER INSTRUCTIONS REFER TO THE RESPONSE BOOKLET. Contents:
More information1 What is the area model for multiplication?
for multiplication represents a lovely way to view the distribution property the real number exhibit. This property is the link between addition and multiplication. 1 1 What is the area model for multiplication?
More informationClassroom Activity to Make Fraction Strips
Classroom Activity to Make Fraction Strips This resource provides instructions on teaching your students how to make fraction strips out of paper. Fraction strips made from copy paper are an excellent
More informationThe candidates are advised that they must always show their working, otherwise they will not be awarded full marks for their answers.
MID SWEDEN UNIVERSITY TFM Examinations 2006 MAAB16 Discrete Mathematics B Duration: 5 hours Date: 7 June 2006 There are EIGHT questions on this paper and you should answer as many as you can in the time
More informationUNC Charlotte Super Competition Comprehensive Test Solutions March 6, 2017
1. Suppose f, g and h are polynomials of degrees 7, 8 and 9 respectively. Let d be the degree of the quotient ( f g) h ( f + g + h), where means composition. Which of the following statements is true?
More informationWhat s In This Book? N Worksheets. L Worksheets. G Worksheets. W Worksheets
Worksheets What s In This Book? This book contains all the worksheets you will need for CSMP for the Upper Primary Grades, Part I. Worksheets are labeled with the same letter and number as the lessons
More informationA tricky node-voltage situation
A tricky node-voltage situation The node-method will always work you can always generate enough equations to determine all of the node voltages. The prescribed method quite well, but there is one situation
More information5-3B Systems Review Puzzle
5-3B Systems Review Puzzle x + y = 4 x y = -2 2x + y = -4 2x + 3y = 4 2x + y = 1 4x 2y = 6 2x + y = 1 x + y = 1 3x 2y = 4-2x + 2y = -1 x = -2y + 1 4 = x + y y = 2 2x x = y 5 y = 4 + 3x 2x + 1 = y x y =
More informationInstructions. Information. Advice
Instructions Use black ink 7C or ball-point pen. Fill in the boxes at the top of this page with your name, centre number and candidate number. Answer all questions. Answer the questions in the spaces provided
More informationMATH 402 : 2017 page 1 of 6
ADMINISTRATION What? Math 40: Enumerative Combinatorics Who? Me: Professor Greg Smith You: students interested in combinatorics When and Where? lectures: slot 00 office hours: Tuesdays at :30 6:30 and
More informationMATH10212 Linear Algebra B Homework 6. Be prepared to answer the following oral questions if asked in the supervision class:
MATH0 Linear Algebra B Homework 6 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra its Applications Pearson, 006 (or other editions) Normally, homework assignments
More informationAlgebra 1 Honors EOC Review #4 Calculator Portion
Algebra 1 Honors EOC Review #4 Calculator Portion 1. Given the data set : 9, 16, 35, 7, 1, 3, 11, 4, 6, 0, 8, 415, 30,, 18, 3, Find the following values : a) Mean b) Median c) Lower Quartile d) Upper Quartile
More informationSolution: There are 30 choices for the first person to leave, 29 for the second, etc. Thus this exodus can occur in. = P (30, 8) ways.
Math-2320 Assignment 7 Solutions Problem 1: (Section 7.1 Exercise 4) There are 30 people in a class learning about permutations. One after another, eight people gradually slip out the back door. In how
More informationTOPIC 12 PROBABILITY SCHEMATIC DIAGRAM
TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of Importance References NCERT Book Vol. II Probability (i) Conditional Probability *** Article 1.2 and 1.2.1 Solved Examples 1 to 6 Q. Nos
More informationCombinatorics and Graph Theory NISER-MA401 Semester 4 of 2010
Combinatorics and Graph Theory NISER-MA401 Semester 4 of 2010 Instructor: Deepak Kumar Dalai 6th January 2010 1. Elementary Enumeration [3, Chapter 1] (a Some basic counting i. Counting the number of strings
More informationUNIT 8: LINEAR FUNCTIONS WEEK 31: Student Packet
Name Period Date UNIT 8: LINEAR FUNCTIONS WEEK 31: Student Packet 31.1 Introduction to Systems of Equations Use variables to write equations and systems of equations. Solve problems involving rate, distance,
More informationUNIT 1 PACKET â PREREQ SKILLS
UNIT 1 PACKET â PREREQ SKILLS manipulations as use of the distributive property, simple factoring, and connecting... Simplifying using Distributive Property and Combining Like Terms:. UNIT 1 Math 621 Simplifying
More informationChapter 3. Independent Events. Introduction
Chapter 3 Independent Events The word probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. Collected Logical
More informationSection 3.3: Linear programming: A geometric approach
Section 3.3: Linear programming: A geometric approach In addition to constraints, linear programming problems usually involve some quantity to maximize or minimize such as profits or costs. The quantity
More informationProf. Thistleton MAT 505 Introduction to Probability Lecture 5
Sections from Text and MIT Video Lecture: Sections 3.3, 3.4, 3.5 http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systems-analysis-andapplied-probability-fall-2010/video-lectures/lecture-2-conditioning-and-bayes-rule/
More information7.1 What is it and why should we care?
Chapter 7 Probability In this section, we go over some simple concepts from probability theory. We integrate these with ideas from formal language theory in the next chapter. 7.1 What is it and why should
More informationPURPLE COMET MATH MEET April 2012 MIDDLE SCHOOL - SOLUTIONS
PURPLE COMET MATH MEET April 2012 MIDDLE SCHOOL - SOLUTIONS Copyright c Titu Andreescu and Jonathan Kane Problem 1 Evaluate 5 4 4 3 3 2 2 1 1 0. Answer: 549 The expression equals 625 64 9 2 1 = 549. Problem
More informationFoundations of Computer Science Lecture 14 Advanced Counting
Foundations of Computer Science Lecture 14 Advanced Counting Sequences with Repetition Union of Overlapping Sets: Inclusion-Exclusion Pigeonhole Principle Last Time To count complex objects, construct
More informationHint 1: a/b is the ratio of a to b. That is a:b Hint 2: RATIO AND PROPORTION SHORTCUTS When two ratios are equal, they are said to be in proportion. If a:b = c:d, then a,b,c & d are proportion. Hint 3:
More information(+ IGCSE) EXAM QUESTION PRACTICE 1. [2 marks] 2. [2 marks]
RATIO [ESTIMATED TIME: 70 minutes] In a school, there is a total of 640 children. The ratio of the number of girls to the number of boys is 7 : 9 How many boys are there in this school? GCSE (+ IGCSE)
More informationReview problems solutions
Review problems solutions Math 3152 December 15, 2017 1. Use the binomial theorem to prove that, for all n 1 and k satisfying 0 k n, ( )( ) { n i ( 1) i k 1 if k n; i k 0 otherwise. ik Solution: Using
More informationChapter 9 Discrete Mathematics
Section 9. Basic Combinatorics 355 Chapter 9 Discrete Mathematics Section 9. Basic Combinatorics Exploration. Six: ABC, ACB, BAC, BCA, CAB, CBA.. Approximately person out of 6, which would mean 0 people
More informationCounting in the Twilight Zone
Chapter 17 Counting in the Twilight Zone In Volume 1 we examined a great many counting methods, but all were based on the rock of common sense. In this chapter we will look at counting methods which go
More informationChapter 12 Answers. Problem of the Week p. 3
Problem of the Week p.. I eat of my chocolate bar, which is equivalent to of her chocolate bar. is equivalent to In decimal form to.. From the decimal form of the fractions, we can see that see that chocolate
More informationNAME: Grade 8 Mathematics Sample Examination Section B. Selected Response. Constructed Response
NAME: Grade 8 Mathematics Sample Examination 2010 Section B Calculator Permitted Selected Response Constructed Response 30 Marks 30 marks You will need a pencil/eraser for this section. You are permitted
More informationPattern Popularity in 132-Avoiding Permutations
Pattern Popularity in 132-Avoiding Permutations The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Rudolph,
More informationMath is Cool Master s
Sponsored by: Zak Designs, Inc. Individual Contest Express all answers as reduced fractions unless stated otherwise. Leave answers in terms of π where applicable. Do not round any answers unless stated
More informationDiscrete mathematics
Discrete mathematics Exercises. Introduction: sets and functions.. Let us consider the sets A = {x N x is even}, B = {x N x > 4}, and C = {x N x < 6}. What are the sets below? B \ C, A \ (B C, B C, (B
More informationMathematics A *P43380A0132* Pearson Edexcel GCSE P43380A. Paper 2 (Calculator) Foundation Tier. Friday 13 June 2014 Morning Time: 1 hour 45 minutes
Write your name here Surname Other names Pearson Edexcel GCSE Centre Number Mathematics A Paper 2 (Calculator) Friday 13 June 2014 Morning Time: 1 hour 45 minutes Candidate Number Foundation Tier Paper
More informationRotational Equilibrium
Rotational Equilibrium In this laboratory, we study the conditions for static equilibrium. Axis Through the Center of Gravity Suspend the meter stick at its center of gravity, with its numbers increasing
More informationF71SM STATISTICAL METHODS
F71SM STATISTICAL METHODS RJG SUMMARY NOTES 2 PROBABILITY 2.1 Introduction A random experiment is an experiment which is repeatable under identical conditions, and for which, at each repetition, the outcome
More information1. A force is a or a. 2. Forces are described by how they are and in what they are going. 3. forces on an object will change the objects motion.
Name period date assigned date due date returned? 1. A force is a or a. 2. Forces are described by how they are and in what they are going. 3. forces on an object will change the objects motion. - - -
More informationSCI-4 Kaechele_Dix_4.2 force,energy,motion Test Exam not valid for Paper Pencil Test Sessions
SCI-4 Kaechele_Dix_4.2 force,energy,motion Test Exam not valid for Paper Pencil Test Sessions [Exam ID:21HKN0 1 Speed is a measure of A motion B force C science D solid 2 A boy helped his neighbors pack
More informationMATH PRIZE FOR GIRLS. Test Version A
Advantage Testing Foundation Ath The Eighth rize For Annual irls MATH PRIZE FOR GIRLS Saturday, September 10, 2016 TEST BOOKLET Test Version A DIRECTIONS 1. Do not open this test until your proctor instructs
More informationLecture Stat 302 Introduction to Probability - Slides 5
Lecture Stat 302 Introduction to Probability - Slides 5 AD Jan. 2010 AD () Jan. 2010 1 / 20 Conditional Probabilities Conditional Probability. Consider an experiment with sample space S. Let E and F be
More informationTest Booklet. Subject: MA, Grade: HS CAHSEE Math Practice Test. Student name:
Test Booklet Subject: MA, Grade: HS CAHSEE Math Practice Test Student name: Author: California District: California Released Tests Printed: Friday December 16, 2011 1 Which number has the greatest absolute
More informationEVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS
EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS CHARLES HELOU AND JAMES A SELLERS Abstract Motivated by a recent work about finite sequences where the n-th term is bounded by n, we evaluate some classes
More informationStatistics 1. Exploring data
Exploring data Chapter assessment 1. George records the time he spends per day surfing the internet for the first three weeks of May. The times, given to the nearest minute, are as follows: 0 6 13 5 18
More informationDiscrete Mathematics: Midterm Test with Answers. Professor Callahan, section (A or B): Name: NetID: 30 multiple choice, 3 points each:
Discrete Mathematics: Midterm Test with Answers Professor Callahan, section (A or B): Name: NetID: 30 multiple choice, 3 points each: 1. If f is defined recursively by: f (0) = -2, f (1) = 1, and for n
More informationMonroe Township School District Monroe Township, New Jersey
Monroe Township School District Monroe Township, New Jersey Preparing for Middle School 6 th Grade *PREPARATION PACKET* **MODIFIED for RESOURCE STUDENTS** Summer 2014 ***SOLVE THESE PROBLEMS WITHOUT THE
More informationMath 140 Introductory Statistics
5. Models of Random Behavior Math 40 Introductory Statistics Professor Silvia Fernández Chapter 5 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. Outcome: Result or answer
More informationQUIZ 1 (CHAPTERS 1-4) SOLUTIONS MATH 119 FALL 2012 KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS
QUIZ 1 (CHAPTERS 1-4) SOLUTIONS MATH 119 FALL 2012 KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% Show all work, simplify as appropriate, and use good form and procedure (as in class). Box in your final
More information