Problem 59 - College Algebra

Question

Explain the Fundamental Counting Principle.

Step-by-Step Solution

Answer

The Fundamental Counting Principle states that if there are multiple ways to do various tasks, then the total number of ways to do all tasks is the product of the number of ways to do each task. An example: a boy with 2 trousers and 3 shirts can make 6 different outfits. The principle can be extended to any number of events.

Step 1: Definition of the Fundamental Counting Principle

The Fundamental Counting Principle states that if there are \(n1\) ways to do something, and \(n2\) ways to do another thing, then there are \(n1 * n2\) ways of doing both. This can be applied to any number of events.

Step 2: Example problem

As an example, consider the situation where a boy has two pairs of trousers and three shirts. To find how many different outfits he can make, apply the counting principle: \( n1 \: (trousers) = 2 \) and \( n2 \: (shirts) = 3 \). Therefore, the total number of outfits is \( n1*n2 = 2*3 = 6 \) outfits.

Step 3: Application of the Principle to Multi-stage Events

The Fundamental Counting Principle can be extended to more than two events. For instance, if there are three events with \( n1, n2 \) and \( n3 \) outcomes respectively, the total number of outcomes for all events would be \( n1*n2*n3 \) and so on.