Fundamental Principle of Counting

Overview

The Fundamental Principle of Counting is a basic concept in combinatorics used to calculate the total number of possible outcomes in a sequence of events. If one event can occur in $n$ ways and another independent event can occur in nn ways, the total number of outcomes for both events is:

$$\text{Total Outcomes} = m \times n$$

This principle can be extended to more than two events. For $k$ events, where the first event can happen in $m_1$ ways, the second in $m_2$ ways, and so on, the total number of outcomes is:

$$\text{Total Outcomes} = m_1 \times m_2 \times \ldots \times m_k$$

Key Ideas

1. Independent Events: The outcomes of one event do not affect the outcomes of another.
2. Systematic Counting: Useful for situations where listing all possibilities is impractical.

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Worked Examples

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Summary

• The Fundamental Principle of Counting calculates total outcomes by multiplying the number of ways each event can occur.
• Formula for $k$ events:

$$\text{Total Outcomes} = m_1 \times m_2 \times \ldots \times m_k$$

• Key applications include calculating combinations of clothing, dice outcomes, and passwords.
• Helps solve problems systematically without listing all possibilities.