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## Chapter 9

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**Chapter 9**Discrete Mathematics**9.1**Basic Combinatorics**What you’ll learn about**• Discrete Versus Continuous • The Importance of Counting • The Multiplication Principle of Counting • Permutations • Combinations • Subsets of an n-Set … and why Counting large sets is easy if you know the correct formula.**Example Using the Multiplication Principle**If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used.**Example Using the Multiplication Principle**If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. You can fill in the first blank 26 ways, the second blank 26 ways, the third blank 26 ways, the fourth blank 26 ways, the fifth blank 10 ways, the sixth blank 10 ways, and the seventh blank 10 ways. By the Multiplication Principle, there are 26×26×26×26×10×10×10 = 456,976,000 possible license plates.**Permutations of an n-Set**There are n! permutations of an n-set.**Example Distinguishable Permutations**Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER.**Example Distinguishable Permutations**Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER. Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations.**Example Counting Combinations**How many 10 person committees can be formed from a group of 20 people?**Example Counting Combinations**How many 10 person committees can be formed from a group of 20 people?**9.2**The Binomial Theorem**What you’ll learn about**• Powers of Binomials • Pascal’s Triangle • The Binomial Theorem • Factorial Identities … and why The Binomial Theorem is a marvelous study in combinatorial patterns.**9.3**Probability**What you’ll learn about**• Sample Spaces and Probability Functions • Determining Probabilities • Venn Diagrams and Tree Diagrams • Conditional Probability • Binomial Distributions … and why Everyone should know how mathematical the “laws of chance” really are.**Probability Distribution for the Sum of Two Fair Dice**Outcome Probability 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36**Example Rolling the Dice**Find the probability of rolling a sum divisible by 4 on a single roll of two fair dice.**Example Rolling the Dice**Find the probability of rolling a sum divisible by 4 on a single roll of two fair dice.**Example Choosing Chocolates**Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?**Example Choosing Chocolates**Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?**Multiplication Principle of Probability**Suppose an event A has probability p1 and an event B has probability p2 under the assumption that A occurs. Then the probability that both A and B occur is p1p2.**Example Choosing Chocolates**Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?**Example Choosing Chocolates**Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?**Example Shooting Free Throws**Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?**Example Shooting Free Throws**Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?**Example Shooting Free Throws**Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?**Example Shooting Free Throws**Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?