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Geometry

Determine all possible arrangements of a set of up to four objects using a tree diagram, or a list in which order is not important.

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After completing this tutorial, you will be able to complete the following:

- Determine all possible arrangements of a set of up to four objects using a tree diagram, or a list in which order is not important.
- Explain what a combination is.
- Explain the difference between permutation and combination.

A tree diagram is a visual display of all possible outcomes in a compound event.

The following is an example of a tree diagram for the possible outcomes of flipping a coin twice.

Possible outcomes are as follows: {heads, heads}, {heads, tails}, {tails, heads}, {tails, tails}, which equals 4 possible outcomes for the event of flipping a coin twice.

A tree diagram is an excellent way to visually introduce possible outcomes to students; students can see how each combination is found by following the "branches" of the tree diagram. However, as the outcomes grow more complex, another way to find a solution is needed, namely the Fundamental Counting Principle.

List of all possible arrangements.

Note that the outcomes from the tree diagram for flipping a coin twice are displayed in brackets. This is a list of all arrangement for the combinations of flipping a coin twice

Ex. {heads, heads}, {heads, tails}, and {tails, tails}, when order doesn't matter.

The list can be shown either vertically or horizontally, since order doesn't matter.

Finding outcomes when order does or doesn't matter.

Using the example of flipping a coin twice, and if the order did matter, the results would be {heads, heads}, {heads, tails}, {tails, heads}, and {tails, tails}, resulting in four outcomes, or combinations

This sometimes can be difficult for students to understand.

In the Activity Object, students will work on an interaction in which they can choose the number of instruments and musicians to make up a band. If you had a band with a guitar and a saxophone, the sound they make will be the same no matter which order they are in. This is a good way to explain to students that {guitar, saxophone} and {saxophone, guitar} are the same combination

If the order does matter when finding outcomes, this is called a permutation.

For example, when selecting the winners of a race, the order matters. The person that finishes first isn't the same as the one who finishes fifth. If the math problem wants to know how many different ways 5 people in a race could finish, you have a permutation.

This Activity Object mentions the idea of permutation in Section 2 as part of the summary. This concept is not explored further here.

The following key vocabulary terms will be used throughout this Activity Object:

- arrangement - the way data is organized or displayed
- combination - an arrangement or listing of data for a given situation when the order isn't important
- outcomes - the possible events of an experiment
- permutation - an arrangement or listing of data for a given situation when the order is important
- tree diagram - a visual map of all the possible outcomes for a given situation

Approximate Time | 15 Minutes |

Pre-requisite Concepts | multiplication of whole numbers, tree diagrams |

Course | Geometry |

Type of Tutorial | Concept Development |

Key Vocabulary | combination, list, tree diagram |