Apply the fundamental counting principle

For a finite multistage counting process, when each completed choice can be followed by a specified number of choices, the total number of outcomes is the product of the numbers of choices at successive stages. Tree diagrams, tables, and organized lists represent this multiplicative structure; the choices need not be probabilistically independent, but varying numbers of later choices require counting branches separately and adding their products. The treatment is limited to finite discrete situations and supports permutations, combinations, and equally likely sample-space calculations without extending to abstract infinite or advanced generalized counting.

Detailed Explanation: Apply the fundamental counting principle

When a process has several stages, and each choice at one stage can be followed by a fixed number of choices at the next stage, multiply the numbers of choices.

\text{Total outcomes} = (\text{choices at stage 1}) (\text{choices at stage 2}) \cdots

Example

A restaurant lets you choose:

3 appetizers
4 main dishes
2 desserts

How many different three-course meals are possible?

Step 1: Identify the stages.

There are three stages:

Choose an appetizer: $3$ choices
Choose a main dish: $4$ choices
Choose a dessert: $2$ choices

Step 2: Multiply the numbers of choices.

For each appetizer, there are $4$ possible main dishes. For each appetizer-and-main-dish combination, there are $2$ possible desserts.

3 \times 4 \times 2 = 24

Answer: There are $\boxed{24}$ different three-course meals.

A tree diagram would show $3$ first branches, $4$ branches from each of those, and $2$ branches from each of those. Multiplying counts all the complete paths through the process.

Learn by doing: Apply the fundamental counting principle

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Probability Sample Space - Definition to Sample Space Count

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