Problem 73 - College Algebra

Question

Explain the Fundamental Counting Principle.

Step-by-Step Solution

Answer

The Fundamental Counting Principle, a method used in probability, states that if there are m ways to do one thing, and n ways to do another, then there are m*n ways to do both. For example, if there are 5 books to choose from and 6 chairs to sit in, there are a total of 30 ways to choose a book and sit in a chair. If there are 3 shirts, 4 pairs of pants, and 2 shoes to choose for an outfit, there are a total of 24 potential outfits.

Step 1: Definition of the Principle

The Fundamental Counting Principle states that if there are m ways to do one thing, and n ways to do another, then there are m*n ways to do both.

Step 2: Application Example 1

Consider a situation in which a student can choose a book from 5 different options and can sit in any of 6 different chairs. Using the Fundamental Counting Principle, we calculate the total number of ways to choose a book and a chair as follows: 5 options for the book * 6 options for the chair = 30 total options. Therefore, there are 30 different ways for the student to choose a book and choose a chair.

Step 3: Application Example 2

For a more complex example, imagine there are 3 shirts, 4 pants, and 2 shoes to choose from an outfit. Using the principle, we calculate: 3 options for the shirt * 4 options for the pants * 2 options for the shoes = 24 total options. So here, there are 24 different outfit combinations the student can put together.