What does the principle of counting express?

• A. There are multiple ways to arrive at a solution
• B. There are m ways to do one thing and n ways to do another, leading to a total of m + n ways to do both
• C. If there are m ways to do one thing and n ways to do another, there are m * n ways to do both
• D. Each unique outcome can be counted only once

Answer: (C)

The principle of counting is foundational in combinatorics and specifically addresses how to determine the total number of outcomes when making multiple selections or choices. In this case, the principle states that if there are ( m ) ways to do one thing and ( n ) ways to do another thing independently, the total number of outcomes when performing both actions is achieved through multiplication, resulting in ( m \times n ) ways to do both.

This multiplication principle stems from the fact that each of the ( m ) outcomes can occur with each of the ( n ) outcomes, thus creating a combination of every available option. For instance, if you are selecting a shirt from 3 options and a pair of pants from 4 options, the overall combinations of outfits you can create is ( 3 \times 4 = 12 ).

Understanding the principle this way allows for efficient calculation of various combinations across different scenarios in probability and counting problems, making it a powerful tool in finite mathematics.