The Fundamental Counting Principle

Rule

Fundamental Counting Principle

If an event A has n possible outcomes and an event B has m possible outcomes, then the total number of different outcomes for A and B combined is n * m. 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. There are n ways to complete the first task and m ways to complete the second task. To complete the whole process, one of the n ways to start it is chosen first. From there, there will be m possible ways to finish the process.

Tree Diagram

This happens for each of the n different ways in which the process can be started. Therefore, there are n* m 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 the notebooks that a store sells.

In this example, the store sells 2 types of notebooks, one with a ring binding and one with a spiral binding. Each notebook type comes in 3 different colors: blue, red, and green. According to the Fundamental Counting Principle, there are 6 different outcomes for which notebook a customer may buy. 2* 3 = 6 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 reflect the actual number of possible outcomes. Returning to the notebooks for sale, suppose now that the spiral-bound notebooks only come in red.

There are still 2 types of notebooks and a total of 3 colors for the ring-bound notebooks. However, the possible number of different notebooks a customer may buy is not 2* 3= 6. Rather, it is 4. This happens because the number of possible colors of the notebook now depends on the type of notebook.

Exercises
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