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Mathematics education began with counting. In the beginning, fingers, beans and buttons were used to help with counting, but these are only practical for small numbers. What happens when a large number of items must be counted?
This chapter focuses on how to use mathematical techniques to count combinations of items.
An important aspect of probability theory is the ability to determine the total number of possible outcomes when multiple events are considered.
For example, what is the total number of possible outcomes when a die is rolled and then a coin is tossed? The roll of a die has six possible outcomes (1, 2, 3, 4, 5 or 6) and the toss of a coin, 2 outcomes (head or tails). Counting the possible outcomes can be tedious.
The simplest method of counting the total number of outcomes is by making a list:
1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T
or drawing up a table:
die | coin |
1 | H |
1 | T |
2 | H |
2 | T |
3 | H |
3 | T |
4 | H |
4 | T |
5 | H |
5 | T |
6 | H |
6 | T |
Both these methods result in 12 possible outcomes, but both these methods have a lot of repetition. Maybe there is a smarter way to write down the result?
One method of eliminating some of the repetition is to use tree diagrams . Tree diagrams are a graphical method of listing all possible combinations of events from a random experiment.
For an integer $n$ , the notation $n!$ (read $n$ factorial) represents:
with the following definition: $0!=1$ .
The factorial notation will be used often in this chapter.
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