Find all possible arrangements of a group of objects using a list or formula in which order is not important.
After completing this tutorial, you will be able to complete the following:
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.
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||counting principle, factorial, concept of permutations|
|Type of Tutorial||Concept Development|
|Key Vocabulary||combinations, counting principle, formula|