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ZingPath: Permutations and Combinations

Combinations and Their Properties

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Permutations and Combinations

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Combinations and Their Properties

Geometry

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

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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