Classroom Resource

12.6 Permutations and Combinations

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

This informational material will introduce three techniques that can be used to count outcomes: (1) Fundamental Counting Principle, (2) permutation, and (3) combinations. It will describe the appropriate situation to use each counting method and describe how to use each method to calculate probabilities of events. The corresponding videos explain each of the three techniques. Practice questions with a PDF answer key are provided.

Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.7

Develop and use the Fundamental Counting Principle for counting independent and dependent events.

Unpacked Content

UP:MA19.FM.7

Vocabulary

  • Fundamental counting principle
  • Independent events
  • Dependent events
  • Tree diagram
  • Branches
  • Node

Knowledge

Students know:

  • How to construct a tree diagram.

Skills

Students are able to:

  • Count the number of events when given a variety of constraints/parameters when the Fundamental Counting Principle can be applied.

Understanding

Students understand that:

  • The Fundamental Counting Principle can be applied in contexts where an ordered list of events occur and there are a ways for the first event to occur, b ways for the second event to occur so the number of ways of the ordered sequence of events occuring is axb.

Body

  1. Develop and use the Fundamental Counting Principle for counting independent and dependent events. a. Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.
    Example: Apply the Fundamental Counting Principle in a context that can be represented by a tree diagram in which there are the same number of branches from each node at each level of the tree.
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.8

Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.

Unpacked Content

UP:MA19.FM.8

Vocabulary

  • Permutations
  • Combinations

Knowledge

Students know:

  • How to use tree diagrams or other counting models .

Skills

Students are able to:

  • Calculate the number of permutations or combinations for a real-world context.

Understanding

Students understand that:

  • Permutation is an ordered selection of r distinct objects from a set of n objects.
  • A combination is a selection of a set of r distinct unordered objects from a set of n objects.

Body

  1. Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.
    Example: If there are r objects chosen from n objects, then the number of permutations can be found by the product [n(n-1) … (n-r)(n-r+1)] as compared to the standard formula n!/(n-r)!. a. Identify differences between applications of combinations and permutations. b. Using application-based problems, calculate the number of permutations of a set with n elements. Calculate the number of permutations of r elements taken from a set of n elements. c. Using application-based problems, calculate the number of subsets of size r that can be chosen from a set of n elements, explaining this number as the number of combinations “n choose r.” d. Using application-based problems, calculate the number of combinations with repetitions of r elements from a set of n elements as “(n + r - 1) choose r.”
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.9

Use various counting techniques to determine probabilities of events.

Unpacked Content

UP:MA19.FM.9

Vocabulary

  • Tree diagrams
  • Combinations
  • Permutations
  • Sample size
  • Independent events
  • Dependent events
  • Mutually exclusive (disjoint) events

Knowledge

Students know:

  • Probability.
  • Permutations and Combinations.
  • Tree diagrams.

Skills

Students are able to:

  • Use a tree diagram or other systematic listing method to determine the number of possible outcomes in an application-based problem.
  • Use combinations and permutations to count the number of possible outcomes in an application -based problem.
  • Determine the probability of an event.

Understanding

Students understand that:

  • Solving probability in a discrete setting requires first applying combinatorial reasoning and counting techniques to determine the size of the event of interest.
  • Some events consist of a sequence of (or partition into) smaller events that may be independent or dependent.

Body

  1. Use various counting techniques to determine probabilities of events.
Link to Resource
https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/12.6/re…

CR Resource Type

Informational Material

Resource Provider

CK-12
Accessibility

Accessibility

Text Resources: Content is organized under headings and subheadings
Video resources: includes closed captioning or subtitles
License

License Type

Custom

Custom License Type (URL)

https://www.ck12info.org/terms-of-use/

Comments or Notes

Students can create a free account online to save their progress.

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