Understanding the Fundamental Counting Principle
In combinatorics, the Fundamental Counting Principle (also known as the Multiplication Rule) is the most basic tool for counting the total number of possible outcomes for a sequence of events.
The Core Concept
If one event can occur in $m$ ways, and a second independent event can occur in $n$ ways, then the total number of ways the two events can occur in sequence is $m \times n$. This rule extends to any number of events: if you have $k$ events with $n_1, n_2, ..., n_k$ options respectively, the total number of combinations is the product of all these counts:
$$\text{Total Outcomes} = n_1 \times n_2 \times ... \times n_k$$
Solving the Election Problem
The Problem: In an election, there are 3 candidates for president, 5 for secretary, and 2 for treasurer. How many possible ways can the election results turn out?
Step-by-Step Solution:
Identify the individual events:
- Event A (Choosing a President): There are 3 options.
- Event B (Choosing a Secretary): There are 5 options.
- Event C (Choosing a Treasurer): There are 2 options.
Apply the Multiplication Rule: Since the choice for each office is independent of the others, we multiply the number of choices for each position together:
$$\text{Total Ways} = 3 \times 5 \times 2$$
Calculate the result: $$3 \times 5 = 15$$ $$15 \times 2 = 30$$
Conclusion: There are 30 different possible outcomes for this election.
Why does this work?
Imagine a tree diagram. For each of the 3 presidential candidates, you can branch out into 5 different secretary choices (creating $3 \times 5 = 15$ branches). From each of those 15 branches, you then branch out into the 2 treasurer options. This confirms that the total number of endpoints is $15 \times 2 = 30$.