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Overview of Permutations and Combinations, Algebra, and Statistics 2018–2019 MATHEMATICSResource Guide Northwest Pa. Collegiate Academy - Erie, PA Overview of Permutations and Combinations, Algebra, and Statistics The vision of the United States Academic Decathlon® is to provide all students the opportunity to excel academically through team competition. Toll Free: 866-511-USAD (8723) • Direct: 712-326-9589 • Fax: 651-389-9144 • Email: [email protected] • Website: www.usad.org This material may not be reproduced or transmitted, in whole or in part, by any means, including but not limited to photocopy, print, electronic, or internet display (public or private sites) or downloading, without prior written permission from USAD. Violators may be prosecuted. Copyright ® 2018 by United States Academic Decathlon®. All rights reserved. Table of Contents INTRODUCTION . 3 2.5 Euler’s Constant . 65 Section 2 Summary: Algebra............71 SECTION 1 OVERVIEW OF PERMUTATIONS AND Section 2 Review Problems: Algebra...... 74 COMBINATIONS................. 4 1.1 The Multiplication Principle . 4 SECTION 3 1.2 Permutations . 8 STATISTICS ......................77 1.3 Combinations . 14 3.1 Descriptive Statistics ................78 3.1.1 Mean, Median, and Mode............78 Section 1 Summary: Overview of Permutations 3.1.2 Range, Quartiles, and IQR............86 and Combinations.....................19 3.2 Measures of Variation . 90 Section 1 Review Problems: Overview of 3.2.1 Variance...........................90 Permutations and Combinations..........20 3.2.2 Standard Deviation..................96 3.2.3 Z-Score ...........................98 SECTION 2 3.3 Basic Probability..................102 ALGEBRA.......................22 3.3.1 Independent Events.................105 2.1 Sequences and Series . 22 3.3.2 Dependent Events................. 110 2.1.1 Arithmetic and Geometric Sequences ...25 3.4 Probability Distributions ............ 115 Northwest Pa. Collegiate Academy - Erie, PA 2.1.2 Arithmetic and Geometric Series .......32 3.4.1 Expected Value ................... 116 2.1.3 Sigma Notation.....................42 3.4.2 Variance and Standard Deviation of 2.2 Polynomials.......................44 Probability Distributions ..................121 2.2.1 Adding and Subtracting Polynomials ...46 3.5 The Binomial Distribution ...........125 2.2.2 Multiplying Polynomials ..............47 3.6 The Normal Distribution............132 2.3 The Binomial Expansion Theorem .....51 Section 3 Summary: Statistics...........137 2.4 Compound Interest.................56 Section 3 Review Problems: Statistics . 141 2.4.1 Investing and Borrowing..............57 2.4.2 Annuities and Loans . .59 CONCLUSION . .148 2018–2019 Mathematics2 Resource Guide Introduction In this year’s Mathematics Resource Guide, we will study three connected areas of mathematics: permutations and combinations, algebra, and statistics. The idea that connections exist between these areas of mathematics may seem strange to you now, but connections and relationships between seemingly disconnected topics such as these are at the heart of mathematics. It is our hope that after reading through this Mathematics Resource Guide, you will be aware of and interested in these connections among different areas of mathematics. The first section of the resource guide will cover permutations and combinations—topics that are sometimes discussed in a general high school math sequence, but perhaps are only given a cursory treatment that often leaves students without a good sense of their power, flexibility, and array of applications. Combinations in particular are extremely important and are used in a wide variety of mathematical contexts, and so a comfort level with these mathematical structures will pay dividends in your future study of mathematics. Section 2 will focus on the topic of algebra. Algebra is a term that has seemingly become synonymous with high school mathematics, and yet the algebra most students learn in high school mathematics courses is only a small fraction of the field of algebra. Our purpose in this section is to highlight some important algebraic ideas and patterns that are often overlooked in a standard study of high school algebra. Being comfortable with arithmetic and geometric sequences and series is at least as important as knowing how to factor a quadratic, but most students spend comparatively little time looking at these sequences and series. Sigma notation for series becomes increasingly important in the study of calculus and mathematics beyond calculus, so this resource guide aims to provide you with plenty of opportunities to practice reading and manipulating sigma notation. The Binomial Expansion Theorem is another foundational topic often left out or deempha- Northwest Pa. Collegiate Academy - Erie, PA sized in traditional high school mathematics, and we will use our previous work with combinations to make sense of this important theorem. Finally, Euler’s constant, e, is often used but rarely properly understood, and so we will look at e both from a contextual and a mathematical perspective. Statistics is a branch of mathematics often only briefly covered in high school mathematics and misunder- stood by much of the general public. In the final section of the Mathematics Resource Guide, we will take a more mathematical look at some of the foundations of statistics, and discuss the reasons different statisti- cal measures, such as mean, median, variance, and standard deviation, were developed. The foundational concepts of probability distributions, the Binomial Distribution, and the Normal Distribution will also be investigated. We hope you find this year’s Mathematics Resource Guide interesting and insightful as we look at connections between these seemingly disparate areas of mathematics. Enjoy! 2018–2019 Mathematics3 Resource Guide Section 1 Section 1 Overview of Permutations Overviewand of Permutations Combinations and Combinations n many high school mathematics sequences, permutations and combinations are often left out all together. When discussed, they are not usually given sufficient time for students to develop an ap- preciation and mastery of these topics. This is certainly not the fault of high school mathematics Iteachers—there is simply too much good content to discuss and not enough time! Yet permutations and combinations are extremely important concepts that underpin a great deal of higher mathematics. Indeed, one of the fields of current mathematical study and research is called combinatorics. Studying permutations and combinations also has great benefits for high school students. Initially these topics are fairly accessible, and some of the early problems can be approached quickly and solved easily. Yet, despite their humble beginnings, they eventually become powerful tools for solving a wide array of problems, and this flexibility makes them very useful. The study of permutations and combinations en- courages and requires visualization, algebraic fluency, and an attention to accurate calculations—all posi- USAD - Santa Ana, CA tive mathematical traits. In last year’s Mathematics Resource Guide, some time was spent discussing permutations and combinations and relating them to probability. As we will use permutations and combinations again in this year’s Math- ematics Resource Guide, we will begin with a refresher on these concepts. If you are able, you can review the portions of last year’s Mathematics Resource Guide on these topics. If not, have no fear! Everything you Northwest Pa. Collegiate Academy - Erie, PA will need to know and understand about permutations and combinations will be discussed in this year’s resource guide. 1.1 THE MULTIPLICATION PRINCIPLE We will begin with a familiar idea from elementary school mathematics: multiplication. Multiplication is most often useful in situations of repeated addition: rather than add up 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3, which could become long and tedious, we instead use multiplication and write 3 · 9 = 27. Many times (hah!) multiplication is introduced as repeated addition, and if asked, many people will give “repeated addition” as a definition, or the definition, of multiplication. (It turns out some mathematicians may dis- 5 agree with this definition, butUSAD that Mathematicsis another story.) Resource Multiplication Guide • 2015-2016 is in some ways the first important mathematical notation encountered, as it represents an operation that is a shortening of a (potentially) long mathResource.indd 5 process of calculation. Let’s start with some straightforward examples. 10/15/2014 10:58:46 PM 2018–2019 Mathematics4 Resource Guide EXAMPLE 1.1A: Toni is having a birthday party next week and wants to give everyone six pieces of candy in their gift bag. If Toni is inviting 15 people to the birthday party, how many pieces of candy does Toni need? SOLUTION: One way to solve this problem is to add up six 15 times: 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6. This makes sense because each of the 15 guests gets six pieces of candy. Although we might expect a first- or second-grader to solve the problem using addition, we are slightly more mathematically sophisticated, and say 15 · 6 = 90. EXAMPLE 1.1B: Toni goes to the store to purchase the candy. Unfortunately, the only candy the store has is little packages of candies with 7 pieces of candy in each package. Toni de- cides to give each guest six packages of this candy. How many pieces of candy does Toni USAD - Santa Ana, CA purchase in order to fill the gift bags? SOLUTION: Toni was planning on getting 90 pieces of candy, since
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