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Basic Combinatorics

Precalculus. Lesson 9.1. Basic Combinatorics. Quick Review. 52. 12. 8. 6. 11. 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

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Basic Combinatorics

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  1. Precalculus Lesson 9.1 Basic Combinatorics

  2. Quick Review 52 12 8 6 11

  3. 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.

  4. Multiplication Principle of Counting

  5. 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 (26 letters in the alphabet) , the second blank 26 ways, the third blank 26 ways, the fourth blank 26 ways, the fifth blank 10 ways (0 thru 9), 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.

  6. Permutations of an n-Set There are n! permutations of an n-set. Permutation: an ordering (ranking) of outcomes in one event. Example: How many ways can a family of five arranged their portraits in a row? 5 • 4 • 3 • 2 • 1 = 5! = 120 Factorial: n! = n(n-1)(n-2)…3•2•1, where 0! = 1

  7. 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! 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320 permutations.

  8. Distinguishable Permutations

  9. Example Distinguishable Permutations Example: How many ways can any two portraits of the family of five be arranged in a row? Two spaces: 5 • 4 = 20

  10. Permutations Counting Formula

  11. Example Permutations Formula Example: How many ways can 8 skaters be awarded gold, silver and bronze medals?

  12. Example Permutations Formula Example: How many ways can the letters in DRESSES be arranged? When an element is repeated, it must be divided out of the total number of arrangements. The number of permutations of n items with r representing elements that are repeated: Number of distinct arrangements:

  13. Combination Counting Formula

  14. Example Counting Combinations How many 10 person committees can be formed from a group of 20 people? Notice that order is not important. There are 184,756 possible committees.

  15. Formula for Counting Subsets of an n-Set

  16. Example Counting Subsets Tony sells one size of pizza, but he claims that his selection of toppings allows for “more than 1000 different choices.” What is the smallest number of toppings Tony could offer? There must be at least 10 toppings to choose from.

  17. Homework: Text pg708/709 Exercises # 2-42 (by 4’s)

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