The Fundamental Counting Principle: Calculating the Total Number of Possible Outcomes in Combinatorics

Fundamental Counting Principle

The Fundamental Counting Principle (FCP) is a fundamental concept in combinatorics that helps us determine the total number of possible outcomes for a series of events or choices

The Fundamental Counting Principle (FCP) is a fundamental concept in combinatorics that helps us determine the total number of possible outcomes for a series of events or choices. It states that if there are ‘m’ ways to do one thing, and ‘n’ ways to do another thing, then there are ‘m x n’ ways to do both things.

To understand the FCP, let’s consider a simple example. Suppose you have 3 shirts (red, blue, and green) and 2 pairs of pants (black and khaki). The FCP allows us to calculate the total number of possible outfits you can create by choosing one shirt and one pair of pants.

Using the FCP, we multiply the number of choices for each item: 3 options for shirts multiplied by 2 options for pants. Therefore, there are 3 x 2 = 6 possible outfits.

We can extend this principle to more complex situations. For example, if we have 3 shirts, 2 pairs of pants, and 4 pairs of shoes, we can use the FCP to find the total number of possible outfits: 3 options for shirts multiplied by 2 options for pants multiplied by 4 options for shoes, resulting in 3 x 2 x 4 = 24 possible outfits.

The FCP can also be applied to scenarios with multiple stages or choices. Let’s say you want to create a 4-digit PIN code. If each digit can be any number from 0 to 9, we can use the FCP to calculate the total number of possible PIN codes. In this case, there are 10 options for each digit (0-9), and since there are four digits, we multiply the possibilities: 10 x 10 x 10 x 10 = 10,000 possible PIN codes.

In summary, the Fundamental Counting Principle is a powerful tool for counting the total number of possible outcomes when we have multiple choices or stages. It states that the total number of outcomes is found by multiplying the number of choices/options for each stage. By understanding and applying this principle, we can analyze and solve various combinatorial problems more effectively.

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