Unit+9

Chapter 9: Discrete Mathematics
In this chapter we have assembled a variety of topics from discrete mathematics, some of which have appeared in traditional precalculus courses (sequences and series, the Binomial Theorem), some of which are common in modern precalculus courses (probability, mathematical induction), and some of which have been touted for years as important for all students, but which have been hard to squeeze into the curriculum (conditional probability, statistics, and data analysis). A new national emphasis on "quantitative literacy" argues powerfully for the inclusion of these topics, even at the cost of omitting some traditional syllabus standards. The hope has been that modern technology can free up some time by re-establishing our pedagogical priorities. We have tried to suggest that possibility throughout this book, even while including some arguably optional topics due to the necessity of adhering to state mandates. Some teachers will assume (perhaps incorrectly) that their students have already seen these discrete topics in their algebra courses. Other teachers will skip this chapter even though they realize that this course is potentially the last opportunity for their students to study these topics during the years of their formal education. If you do wind up skipping this chapter in your precalculus course, you might consider familiarizing yourself with the material on your own. You may not use it in calculus, but you will see it in a multitude of other settings.





Section 9.1 Basic Combinatorics
**Objectives**

You will be able to use the multiplication principle of counting, permutations, or combinations to count the number of ways that a described task can be done. **Key Ideas**
 * Combination || Multiplication principle of counting ||
 * Continuum || //n//-set ||
 * Discrete mathematics || Permutation ||
 * Explanatory variable || Response variable ||

Section 9.2 The Binomial Theorem
**Objectives**

You will be able to expand an integral power of a binomial using the Binomial Theorem or Pascal's triangle. You will also be able to find the coefficient of a given term of a binomial expansion. Binomial coefficientBinomial TheoremPascal's triangle
 * Key Ideas**

Section 9.3 Probability
**Objectives**

You will be able to identify a sample space and calculate probabilities and conditional probabilities of events in sample spaces with equally likely or unequally likely outcomes. **Key Ideas**
 * Binomial distribution || Probability distribution ||
 * Conditional probability || Probability function ||
 * Equally likely outcomes || Probability of an event ||
 * Event || Sample space ||
 * Independent event || Tree diagram ||
 * Multiplication principle of probability || Venn diagram ||

Section 9.4 and 9.5 Sequences and Series
**Objectives**

You will be able to express arithmetic and geometric sequences explicitly and recursively, including in sigma notation. You will be able to use basic summation formulas to find the sums of finite series or of convergent infinite geometric series. **Key Ideas**
 * Arithmetic sequence || Infinite series ||
 * Convergent series || Partial sum (of a series) ||
 * Divergent series || Recursively-defined sequence ||
 * Fibonacci sequence || Sequence ||
 * Geometric sequence || Sum of an infinite series ||
 * Index of summation || Summation notation ||

Section 9.6 Mathematical Induction
**Objectives**

You will be able to understand proofs by mathematical induction and will be able to produce mathematical induction proofs for simple propositions. **Key Ideas**
 * Anchor || Inductive step ||
 * Inductive hypothesis || Mathematical induction ||