Systematic Counting – Listing Without Missing or Repeating
Systematic counting means listing every possible outcome in an organised way, so that nothing is accidentally missed and nothing is accidentally counted twice. Rather than listing outcomes randomly, a systematic approach fixes one part first (like the first digit of a code), then works through every option for the remaining parts in order.
This habit of organisation is what makes larger counting problems manageable, and it's the direct precursor to the formulas you'll meet later in this section – every one of those formulas is really just a shortcut for a systematic list you could, in principle, still write out by hand.
Listing Systematically
Fix the first choice, then systematically list every option for the remaining choices before moving to the next first choice.
Fixing the first digit as 1: 12, 13. Fixing it as 2: 21, 23. Fixing it as 3: 31, 32. That's 6 numbers in total.
Real-Life Application
- Quality control: systematically testing every combination of settings to find a fault.
- Puzzle solving: systematically working through possibilities in logic puzzles like Sudoku.
- Recipe testing: systematically trying every ingredient combination in a taste test.
Key Takeaways
- Systematic counting organises a list so nothing is missed or double-counted.
- A common strategy is to fix one choice and list all options for the rest, then repeat.
- This approach is the foundation for the counting formulas covered later in this section.
Practice: Systematic Counting
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