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Rule

# Fundamental Counting Principle

If an event has possible outcomes and an event has possible outcomes, then the total number of different outcomes for and combined is This principle is used to find the number of possible outcomes for a combination of independent events.

### Proof

Informal Justification
Consider an arbitrary process that can be divided into two tasks. Now imagine that there are different ways of completing the first task and different ways of completing the second task. To complete the whole process, first, one of the ways to start it should be chosen. Then, there will be possible ways to finish it.
This happens for each of the different ways in which the process can be started. Therefore, there are different ways of completing the process. This is a generic argument that can be applied in multiple scenarios. For example, the following diagram shows the different choices of notebooks that a store sells.
In this example, the store sells types of notebooks — one with a spiral binding and one without. Each notebook type comes in different colors — blue, red, and green. According to the Fundamental Counting Principle, there are different outcomes for what notebook a customer may buy.
It should be noted that this is an informal justification and should not be taken as a formal proof.

### Extra

Counting the Outcomes of Dependent Events

As mentioned above, this principle holds true only if the events are independent of each other. If the events are dependent, multiplying the number of possible outcomes for each event will not be correct. Considering the previous example, suppose now that the spiral-bound notebooks came only in red.

There are still types of notebooks and a total of colors for the non spiral bound notebooks. However, the possible number of different notebooks a customer may buy is not Rather, it is This happens because, in this case, the possible colors for a notebook depend on the type of notebook.