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2. Using Boundless Presentations Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: • The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. http://boundless.com/teaching-platform • Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. If you have any questions or problems please email: educators@boundless.com Free to share, print, make copies and changes. Get yours at www.boundless.com

3. About Boundless • Boundless is an innovative technology company making education more affordable and accessible for students everywhere. The company creates the world’s best open educational content in 20+ subjects that align to more than 1,000 popular college textbooks. Boundless integrates learning technology into all its premium books to help students study more efficiently at a fraction of the cost of traditional textbooks. The company also empowers educators to engage their students more effectively through customizable books and intuitive teaching tools as part of the Boundless Teaching Platform. More than 2 million learners access Boundless free and premium content each month across the company’s wide distribution platforms, including its website, iOS apps, Kindle books, and iBooks. To get started learning or teaching with Boundless, visit boundless.com. Free to share, print, make copies and changes. Get yours at www.boundless.com

4. Combinatorics Combinatorics and Probability The Binomial Theorem Probability ] Combinatorics and Probability Free to share, print, make copies and changes. Get yours at www.boundless.com

5. Combinatorics and Probability > Combinatorics Combinatorics • Counting Rules and Techniques • Permutations • Permutations of Distinguishable Objects • Permutations of Nondistinguishable Objects • Combinations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/combinatorics-and-probability-343/combinatorics-57/

6. Combinatorics and Probability > The Binomial Theorem The Binomial Theorem • Binomial Expansions and Pascal's Triangle • Binomial Expansion and Factorial Notation • Finding a Specific Term Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/combinatorics-and-probability-343/the-binomial-theorem-58/

7. Combinatorics and Probability > Probability Probability • Fundamentals of Probability • Unions and Intersections • Conditional Probability • Complementary Events • The Addition Rule • The Multiplication Rule • Independence • Experimental Probabilities Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/combinatorics-and-probability-343/probability-255/

8. Appendix Free to share, print, make copies and changes. Get yours at www.boundless.com

9. Combinatorics and Probability Key terms • binomialA polynomial consisting of two terms, or monomials, separated by an addition or subtraction symbol. • binomial coefficientA coefficient of any of the terms in the expansion of the binomial power . • binomial coefficientA coefficient of any of the terms in the expansion of the binomial power • binomial coefficientA coefficient of any of the terms in the expansion of the binomial power . • binomial distributionThe discrete probability distribution of the number of successes in a sequence of independent yes/no experiments, each of which yields success with probability . • combinationA way of selecting elements from a set, where order does not matter. • combinatoricsA branch of mathematics that studies (usually finite) collections of objects that satisfy specified criteria. • conditional probabilityThe probability that an event will take place given the restrictive assumption that another event has taken place, or that a combination of other events has taken place • discreteSeparate; distinct; individual; non-continuous. • disjointHaving no members in common; having an intersection equal to the empty set. • eventA subset of the sample space. • exhaustiveincluding every possible element Free to share, print, make copies and changes. Get yours at www.boundless.com

10. Combinatorics and Probability • experimentSomething that is done that produces measurable results, called outcomes. • experimental probabilityThe probability that a certain outcome will occur, as determined through experiment. • factorialThe result of multiplying a given number of consecutive integers from to the given number. In equations, it is symbolized by an exclamation mark (). For example, . • factorialThe result of multiplying a given number of consecutive integers from to the given number. In equations, it is symbolized by an exclamation mark (). For example, . • factorialThe result of multiplying a given number of consecutive integers from to the given number. In equations, it is symbolized by an exclamation mark (). For example, . • independenceThe occurrence of one event does not affect the probability of the occurrence of another. • independentNot dependent; not contingent or depending on something else; free. • independentNot contingent or dependent on something else. • integerAn element of the infinite and numerable set • integerAn element of the infinite and numerable set . • multiplicityThe number of values for which a given condition holds. • mutually exclusivedescribing multiple events or states of being such that the occurrence of any one implies the non-occurrence of all the others Free to share, print, make copies and changes. Get yours at www.boundless.com

11. Combinatorics and Probability • outcomeOne of the individual results that can occur in an experiment. • permutationAn ordering of a finite set of distinct elements. • permutationAn ordering of a finite set of distinct elements. • permutationAn ordering of a finite set of distinct elements. • polynomialAn expression consisting of a sum of a finite number of terms: each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power. • probabilityThe relative likelihood of an event happening. • probability theoryThe mathematical study of probability (the likelihood of occurrence of random events in order to predict the behavior of defined systems). • sample spaceThe set of all possible outcomes of a game, experiment or other situation. • sample spaceThe set of all outcomes of an experiment. Free to share, print, make copies and changes. Get yours at www.boundless.com

12. Combinatorics and Probability Coin Flip Often in sports games, such as tennis, a coin flip is used to determine who will serve first because heads and tails are complementary events. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia."Heads or tails."CC BY-SAhttp://commons.wikimedia.org/wiki/File:Heads_or_tails.jpgView on Boundless.com

13. Combinatorics and Probability Die Dice are often used when learning the rules of probability. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Nuvola apps atlantik."CC BY-SAhttp://en.wikipedia.org/wiki/File:Nuvola_apps_atlantik.pngView on Boundless.com

14. Combinatorics and Probability One stack of cards in a game of solitaire To find out how many possible combinations of cards there are below the seven of spades, we use the concept of permutations to calculate the possible arrangements of cards. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX."Kenny Felder, Advanced Algebra II: Conceptual Explanations. December 11, 2012."CC BY 3.0http://cnx.org/content/m19072/latest/?collection=col10624View on Boundless.com

15. Combinatorics and Probability The 6 permutations of 3 balls If one has three different colored balls, there are six distinct ways to order them, as shown. These six distinct orderings are as follows: red-green-blue, red-blue-green, green-red-blue, green-blue-red, blue-red-green, and blue-green-red. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Permutations RGB."CC BY-SAhttp://en.wikipedia.org/wiki/File:Permutations_RGB.svgView on Boundless.com

16. Combinatorics and Probability Union of Two Sets The shaded Venn Diagram shows the union of set (the circle on left) with set (the circle on the right). It can be written shorthand as . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Venn0111."CC BY-SAhttp://en.wikipedia.org/wiki/File:Venn0111.svgView on Boundless.com

17. Combinatorics and Probability Intersection of Two Sets Set is the circle on the left, set is the circle on the right, and the intersection of and , or , is the shaded portion in the middle. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Venn0001."CC BY-SAhttp://en.wikipedia.org/wiki/File:Venn0001.svgView on Boundless.com

18. Combinatorics and Probability Pascal's Triangle Pascal's triangle with 5 rows. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Pascal's_triangle_5.svg."CC BY-SA 3.0https://en.wikipedia.org/wiki/Pascal's_triangleView on Boundless.com

19. Combinatorics and Probability Pascal's Triangle Each number in the triangle is the sum of the two directly above it. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Pascal's triangle."GNU FDLhttp://en.wikipedia.org/wiki/Pascal's_triangleView on Boundless.com

20. Combinatorics and Probability Independent Events Selecting two cards from a deck by first selecting one, then replacing it in the deck before selecting a second is an example of independent events. Free to share, print, make copies and changes. Get yours at www.boundless.com Flickr."All sizes | Ace of Spades Card Deck Trick Magic Macro 10-19-09 2 | Flickr - Photo Sharing!."CC BYhttp://www.flickr.com/photos/stevendepolo/4028160820/sizes/o/in/photostream/View on Boundless.com

21. Combinatorics and Probability Attribution • Wikipedia."binomial distribution."CC BY-SA 3.0https://en.wikipedia.org/wiki/Binomial_distribution • Wikipedia."Empirical probability."CC BY-SA 3.0http://en.wikipedia.org/wiki/Empirical_probability • Wiktionary."discrete."CC BY-SA 3.0https://en.wiktionary.org/wiki/discrete • Wiktionary."experimental probability."CC BY-SA 3.0http://en.wiktionary.org/wiki/experimental_probability • Wikipedia."Probability axioms."CC BY-SA 3.0http://en.wikipedia.org/wiki/Probability_axioms • Wiktionary."sample space."CC BY-SA 3.0http://en.wiktionary.org/wiki/sample_space • Wiktionary."disjoint."CC BY-SA 3.0http://en.wiktionary.org/wiki/disjoint • Wiktionary."independent."CC BY-SA 3.0http://en.wiktionary.org/wiki/independent • Wikipedia."Bayes' theorem."CC BY-SA 3.0http://en.wikipedia.org/wiki/Bayes'_theorem • Wikipedia."Conditional probability."CC BY-SA 3.0http://en.wikipedia.org/wiki/Conditional_probability • Wiktionary."conditional probability."CC BY-SA 3.0http://en.wiktionary.org/wiki/conditional_probability • Wiktionary."independent."CC BY-SA 3.0http://en.wiktionary.org/wiki/independent • Wikipedia."Union (set theory)."CC BY-SA 3.0http://en.wikipedia.org/wiki/Union_(set_theory) • Wiktionary."disjoint."CC BY-SA 3.0http://en.wiktionary.org/wiki/disjoint • Wikipedia."Intersection (set theory)."CC BY-SA 3.0http://en.wikipedia.org/wiki/Intersection_(set_theory) • Wiktionary."independent."CC BY-SA 3.0http://en.wiktionary.org/wiki/independent • Wikipedia."Complementary event."CC BY-SA 3.0http://en.wikipedia.org/wiki/Complementary_event Free to share, print, make copies and changes. Get yours at www.boundless.com

22. Combinatorics and Probability • Wiktionary."mutually exclusive."CC BY-SA 3.0http://en.wiktionary.org/wiki/mutually_exclusive • Wiktionary."exhaustive."CC BY-SA 3.0http://en.wiktionary.org/wiki/exhaustive • Wikipedia."Probability axioms."CC BY-SA 3.0http://en.wikipedia.org/wiki/Probability_axioms • Wikipedia."addition rule."CC BY-SA 3.0http://en.wikipedia.org/wiki/addition%20rule • Wiktionary."probability."CC BY-SA 3.0http://en.wiktionary.org/wiki/probability • Wikipedia."Probability."CC BY-SA 3.0https://en.wikipedia.org/wiki/Probability • Wikidot."Some rules of probability - Statistics."CC BY-SAhttp://statistics.wikidot.com/ch5 • Wiktionary."sample space."CC BY-SA 3.0http://en.wiktionary.org/wiki/sample_space • Boundless Learning."Boundless."CC BY-SA 3.0http://www.boundless.com//statistics/definition/multiplication-rule • Wikidot."Some rules of probability - Statistics."CC BY-SAhttp://statistics.wikidot.com/ch5 • Wikipedia."Independence (probability theory)."CC BY-SA 3.0http://en.wikipedia.org/wiki/Independence_(probability_theory) • Wikipedia."independence."CC BY-SA 3.0http://en.wikipedia.org/wiki/independence • Wiktionary."probability theory."CC BY-SA 3.0http://en.wiktionary.org/wiki/probability_theory • OpenStax CNX."Roberta Bloom, Probability Topics: Independent & Mutually Exclusive Events (modified R. Bloom). September 17, 2013."CC BY 3.0http://cnx.org/content/m18868/latest/ • Wikipedia."Combinatorics."CC BY-SA 3.0http://en.wikipedia.org/wiki/Combinatorics • Wikipedi."Pigeonhole principle."CC BY-SA 3.0https://en.wikipedia.org/wiki/Pigeonhole_principle • Wikipedia."Bijective proof."CC BY-SA 3.0http://en.wikipedia.org/wiki/Bijective_proof • Wikipedia."Double counting (proof technique)."CC BY-SA 3.0http://en.wikipedia.org/wiki/Double_counting_(proof_technique) • Wikipedia."Bijection."CC BY-SA 3.0https://en.wikipedia.org/wiki/Bijection Free to share, print, make copies and changes. Get yours at www.boundless.com

23. Combinatorics and Probability • Wikipedia."Inclusion-exclusion principle."CC BY-SA 3.0https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle • Wiktionary."polynomial."CC BY-SA 3.0http://en.wiktionary.org/wiki/polynomial • Wiktionary."combinatorics."CC BY-SA 3.0http://en.wiktionary.org/wiki/combinatorics • Wikipedia."Combinatorial principles."CC BY-SA 3.0http://en.wikipedia.org/wiki/Combinatorial_principles • Wikipedia."Permutation."CC BY-SA 3.0http://en.wikipedia.org/wiki/Permutation • Wiktionary."permutation."CC BY-SA 3.0http://en.wiktionary.org/wiki/permutation • Wiktionary."factorial."CC BY-SA 3.0http://en.wiktionary.org/wiki/factorial • OpenStax CNX."Kenny Felder, Advanced Algebra II: Conceptual Explanations. September 17, 2013."CC BY 3.0http://cnx.org/content/m19072/latest/?collection=col10624 • Wikipedia."Combination."CC BY-SA 3.0http://en.wikipedia.org/wiki/Combination • Wiktionary."binomial coefficient."CC BY-SA 3.0http://en.wiktionary.org/wiki/binomial_coefficient • Wikipedia."Permutation."CC BY-SA 3.0https://en.wikipedia.org/wiki/Permutation • OpenStax CNX."Kenny Felder, Advanced Algebra II: Conceptual Explanations. September 17, 2013."CC BY 3.0http://cnx.org/content/m19071/latest/?collection=col10624 • Wikipedia."Binomial theorem."CC BY-SA 3.0http://en.wikipedia.org/wiki/Binomial_theorem • Wiktionary."integer."CC BY-SA 3.0http://en.wiktionary.org/wiki/integer • Wikipedia."Permutation."CC BY-SA 3.0http://en.wikipedia.org/wiki/Permutation • Wikipedia."Factorial."CC BY-SA 3.0http://en.wikipedia.org/wiki/Factorial • Wiktionary."permutation."CC BY-SA 3.0http://en.wiktionary.org/wiki/permutation • Wiktionary."factorial."CC BY-SA 3.0http://en.wiktionary.org/wiki/factorial • Wikipedia."Permutation."CC BY-SA 3.0http://en.wikipedia.org/wiki/Permutation Free to share, print, make copies and changes. Get yours at www.boundless.com

24. Combinatorics and Probability • Wiktionary."permutation."CC BY-SA 3.0http://en.wiktionary.org/wiki/permutation • Wiktionary."multiplicity."CC BY-SA 3.0http://en.wiktionary.org/wiki/multiplicity • Wikipedia."Pascal's triangle."CC BY-SA 3.0http://en.wikipedia.org/wiki/Pascal's_triangle • Wikipedia."Binomial theorem."CC BY-SA 3.0http://en.wikipedia.org/wiki/Binomial_theorem • Wiktionary."binomial."CC BY-SA 3.0http://en.wiktionary.org/wiki/binomial • Wikipedia."Binomial theorem."CC BY-SA 3.0http://en.wikipedia.org/wiki/Binomial_theorem • Wikipedia."Factorial."CC BY-SA 3.0http://en.wikipedia.org/wiki/Factorial • Wiktionary."factorial."CC BY-SA 3.0http://en.wiktionary.org/wiki/factorial Free to share, print, make copies and changes. Get yours at www.boundless.com