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Comprehensive Guide to Permutation Investigations and Analysis

Explore the fundamentals of permutations and investigate various scenarios using factorials, repetitions, and circular permutations. Discover patterns, formulas, and additional counting methods in this comprehensive study.

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Comprehensive Guide to Permutation Investigations and Analysis

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  1. Permutation Investigations Honors Analysis

  2. Ft. Thomas Exchange

  3. Section 1.1 VocabularY • Independent Events: Events that do not affect each other • Sample Space: List of all possible outcomes • Fundamental Counting Principle: If events are independent, the number of ways both can occur is the product of the number possible outcomes for each event. • Permutation: An arrangement in which ORDER MATTERS

  4. Permutations • In how many unique orders can 5 people sit in a row of chairs? • In how many orders can the letters of the word “pencil” be arranged? • In how many ways can five people place first, second, and third in a race?

  5. Permutation Investigations 5P3 is a notation for writing the permutation of 5 items taken 3 at a time. 5P3 6P4 = 8P2= How might you calculate these values using only factorials?? Use your pattern to write a formula for the permutation of n items taken r at a time: nPr

  6. Permutations with Repetition • The goal: Find the number of possible arrangements in the word MISSISSIPPI

  7. Permutations with Repetition • Calculate the following factorial values: 0! = 1! = 2! = 3! = 4! = 5! = 6! = 7! =

  8. Permutations with Repetition Determine the number of unique arrangements created by the following sets of letters and complete the chart: Do you notice any patterns? **Hint: Factorials!

  9. Permutations with Repetition • How many arrangements could be created from the letters in “cheese” if only the e’s can be moved and are distinct from each other (try using e1 e2 e3) CH____S__ How might this help you develop a formula?

  10. Circular Permutations • In how many orders can you sit 3 people around a table? (Careful – no “beginning point”) • 4 people? • 5 people? • Do you notice a pattern?

  11. Additional Counting Methods The chart below shows information about Math Club members: In how many ways could a boy OR girl be selected? In how many ways could a sophomore OR a girl be selected? If order matters, in how many ways could two boys be selected? (A boy AND a boy)

  12. Vocabulary • Events are mutually exclusive if one excludes the other from happening. n(A or B) = n(A) + n(B) • If events are non-mutually exclusive, be careful to subtract the number of ways the events can overlap! n(A or B) = n(A) + n(B) – n(A ∩ B) • Multiple Dependent Events: n(A and B) = n(A) · n(B | A), where n(B | A) is the number of ways event B can occur after A occurs

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