Combinatorics Word Problems – Real-World Applications
Everything covered in this section – the Fundamental Counting Principle, factorials, permutations, and combinations – comes together in real-world combinatorics word problems. The single most important skill for these problems is deciding, right at the start, whether the situation calls for a permutation (order matters) or a combination (order doesn't matter).
A reliable test: if swapping two of your chosen items would create a genuinely different outcome, it's a permutation. If it would describe exactly the same outcome, it's a combination.
Solving Combinatorics Word Problems
Decide first whether order matters (permutation) or not (combination), then apply the matching formula with the correct n and r.
Order matters (1st is different from 2nd): 10P3 = 10 × 9 × 8 = 720.
Order doesn't matter for a team: 12C5 = 792.
Real-Life Application
- Sports: counting possible race results, team lineups, or tournament brackets.
- Business: counting possible product combinations or committee formations.
- Technology: counting possible passwords, network configurations, or test cases.
Key Takeaways
- The first step in any word problem is deciding: permutation or combination?
- If order matters, use permutations; if it doesn't, use combinations.
- This applies every formula from this section to real-world decision-making.
Practice: Combinatorics Word Problems
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