Different Strategies

Overview

When solving problems in counting and probability, selecting the right strategy is crucial for accuracy and efficiency. The Different Strategies note focuses on approaches to systematically organise and calculate outcomes for a variety of counting problems. These strategies include:

Listing Outcomes

• List all possible outcomes for simple problems.
• Useful for visualising and verifying possibilities in small sample spaces.

Using the Fundamental Principle of Counting

Multiply the number of choices at each step to find the total number of outcomes.

Factorials (n!n!)

• Used to count arrangements (permutations) of $n$ distinct objects.
• Formula:

$$n! = n \times (n-1) \times \dots \times 1$$

Combinations and Permutations

Permutations:

Arrangements where order matters.

Formula:

$$P(n, r) = \frac{n!}{(n-r)!}$$

Combinations:

Selections where order does not matter.

Formula:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

Using Venn Diagrams

• A visual way to represent sets and their intersections.
• Useful for solving problems involving multiple overlapping groups.

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Worked Examples

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Summary

• Key Strategies:
  1. Listing outcomes.
  2. Fundamental Principle of Counting: Multiply choices.
  3. Factorials: Use $n!$ for arranging objects.
  4. Combinations ($C(n, r)$): Selections where order doesn't matter.
  5. Permutations ($P(n, r)$): Arrangements where order matters.
  6. Venn diagrams: Visualise and solve set problems.
• These strategies are essential for solving problems in counting and probability systematically.