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# ZingPath: Permutations and Combinations

## Combinations and Their Properties                       Searching for

## Permutations and Combinations

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### Lesson Focus

#### Combinations and Their Properties

Geometry

Find all possible arrangements of a group of objects using a list or formula in which order is not important.

### Now You Know

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

• Determine all possible arrangements of a group of objects by making lists in which order is not important.
• Explain what a combination is.
• Explain the difference between permutation and combination.
• Apply the combination formula to solve problems.

### Everything You'll Have Covered

A factorial is the product of all positive integers up to and including a given integer. It is denoted by an exclamation mark ( ). The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. A combination is an arrangement of objects in which order does not matter

An arrangement of r objects, without regard to order and without repetition, selected from n distinct objects is called a combination of n objects taken r at a time and is denoted as: where n is the number of objects available for selection and r is the number of objects to be selected Combinations can also be notated as: In combinations, we count groups where order is not important.

For example, in a conference of 9 schools, how many intraconference football games are played during the season if the teams all play each other exactly once?

When the teams play each other, we're counting match-ups, so order doesn't matter. For each game there is a group of two teams playing. Team A playing Team B is the same game as Team B playing Team A

First, find n and r : n is the number of teams we have to choose from, which is 9, and r is the number of teams we are using at a time, which is 2. Substitute these numbers into the combination formula and simplify: This means there are 36 different games in the conference.

If we don't select any element from n elements, the combination of this selection is 1. If we select n elements from n elements, the combination of this selection is also 1 Whenever we select an element from among n elements, the combination of this selection is n. Selecting r elements from n elements is the same as selecting (n?r) elements from the same set. ### Tutorial Details

 Approximate Time 20 Minutes Pre-requisite Concepts counting principle, factorial, concept of permutations Course Geometry Type of Tutorial Concept Development Key Vocabulary combinations, counting principle, formula