Systematic lists and the product rule · Higher
| English | 中文 | Pinyin |
|---|---|---|
| systematic list/ˌsɪstəˈmætɪk lɪst/ | 系统列表 | xì tǒng liè biǎo |
Have we listed every choice?
- A cafe offers tea or juice with apple, banana or melon. How can a list prove that no lunch choice was missed?
- This lesson studies systematic list 系统列表: A list made in a fixed order so every allowed outcome appears exactly once.
Choose the mathematical structure
- Fix one first choice and list every permitted second choice before moving to the next first choice. A table gives the same structure. When every one of m first choices allows n second choices, the Higher product rule gives mn outcomes. If restrictions change the options, count the allowed rows separately.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines systematic list?
A list made in a fixed order so every allowed outcome appears exactly once.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
The list is tea-apple, tea-banana, tea-melon, juice-apple, juice-banana, juice-melon: six outcomes. If tea-melon is unavailable, five remain. For a three-digit code with digits 1,2,3 and no repeats, choose 3 then 2 then 1 possibilities, giving six codes: 123,132,213,231,312,321.
Systematic lists and the product rule
Fix one first choice and list every permitted second choice before moving to the next first choice
Compare the model with the worked case and explain one change.
How many cafe choices are in the complete list?
There are three listed foods under each of two drinks: six listed outcomes.
Test a tempting shortcut
- State whether order matters and whether repetition is allowed. Choosing A then B can be different from B then A for a code, but not for an unordered two-person team. A restriction can make a simple product invalid.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Three available first choices and four second choices always give twelve outcomes even if some combinations are forbidden. This claim is false. Explain which definition or assumption it violates.
How many remain when tea-melon is removed?
Remove one of the six outcomes: five remain.
Three available first choices and four second choices always give twelve outcomes even if some combinations are forbidden.
State whether order matters and whether repetition is allowed. Choosing A then B can be different from B then A for a code, but not for an unordered two-person team. A restriction can make a simple product invalid.
Interpret a new situation
- Foundation should justify answers with a complete ordered list or grid. Higher may compress a verified structure using the product rule, but must still check restrictions.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many three-digit codes use 1,2,3 once each?
List 123,132,213,231,312,321: six codes.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.1. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A list made in a fixed order so every allowed outcome appears exactly once. Choose the relationship, show the method, check its assumptions and interpret the result.