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Searching for ## Permutations and Combinations

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Explore the full path to learning Permutations and Combinations

Geometry

Learners will identify and calculate the number of permutations with repetition.

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

- Explain that all permutations of a group of identical objects are the same.
- Calculate the number of permutations with repetition.

All permutations of a group of identical objects are the same.

Let's use the example of planting three tulips in a box. If all three objects are different, i.e. they have different colors, then there would be 3! (read as "three factorial", or 3 × 2 × 1 = 6) possible arrangements for the tulips.

But if all of the objects are identical, e.g. they are all yellow, then there is no way to tell them apart no matter what order they are planted in. Thus, the number of permutations of the group of three identical yellow tulips would be a single permutation.

The formula used for permutations with repetition is .

Let's take our example and expand on it. Let's say we want to find out the number of different ways that we can plant two identical yellow tulips and three identical red tulips in a flower box.

There are a total of 5! (5 × 4 × 3 × 2 × 1 = 120) possible arrangements of the five tulips. However, since there are two yellow and three red that are identical, we apply the above formula:

There is a total of 10 different ways to plant the five tulips (with two identical yellow tulips and three identical red) in a flower box. This is because the identical ones can be changed around in the order with no difference, because they are identical.

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

- counting principle - a method used to calculate all of the possible combinations of a given number of events; the formula is , where is the number of possibilities of event 1, is the number of possibilities of event 2, etc., and is the product of all of these event possibilities multiplied together
- factorial - the product of all the integers up to and including a given integer. It is denoted with the exclamation mark (!)
- permutation - all possible arrangements of a collection of things
- permutations with repetition - all possible arrangements of a collection of things with some repeated items
- possible arrangements - the total number of ways that a collection can be rendered

Approximate Time | 25 Minutes |

Pre-requisite Concepts | Permutations |

Course | Geometry |

Type of Tutorial | Concept Development |

Key Vocabulary | permutation, permutation with repetition, repetition |