Counting problems could be addressed a series of decisions where every decision involves choosing from more than one options. In several cases, the possible outcome could be even found by multiplying the number of options for the first decision times the available number of options for second decision, third decision and so on.
The fundamental counting principle
According to the fundamental counting principle, the product of the finite number of options available is likely to be the number of possible arrangements. The fundamental principle of counting could be expressed within a three-step process.
In more technical terms, the fundamental counting principle could be expressed as a rule that allows counting the number of total possible outcomes within a situation. It states that there could be ways of doing something and ways of doing the other. Thus, there would be ways of doings both of performing both the actions.
Now, consider several examples for the fundamental counting principle.
Example -1: Suppose that there are several slots representing course meals that you may be ordering. For any 6 courses you would have 3 choices for appetizers, 2 for soups and 4 for salads. Also, with 5 choices of main course, you will have choices from 10 beverages and 3 desserts. To know how many unique six course meals you can have, list down the number of choices and multiply.
3 x 2 x 4 x 5 x 10 x 3 = 3,600. Which means you may avail 3,600 unique 6-course meals from the available options.
Example -2: a girl is trying to make out what to wear. She has several shirts of the colors: purple, red and blue. Also, she has pants in the colors: black and white. Assuming that she’s already picked up one of the shirts; makes out how many different outfits can she choose from.
We already know what the fundamental counting principle states. “If there are ways of doing something (choosing a shirt in this case), and ways to do the other thing (like choosing pants here), then there would be total combinations to choose from. In the case mentioned above, there are total 3 options for choosing a shirt and 2 options for choosing pants Thus, there would be 3 x 2 i.e. 6 different pairs of dresses to choose from.
Based on the above considerations for the fundamental principle of counting, probable decisions could be made out.