The decision and its conditions
The same people can be chosen with or without distinct roles. Assigning a chair and secretary counts ordered pairs; choosing an unordered pair counts each pair twice in that ordered count, so divide by two. A product rule counts combinations of independent choices, but forbidden combinations must be removed explicitly. For overlapping events, adding category counts double-counts their intersection. Subtract that overlap once before forming the probability of either event. State the sample space and whether order matters before using any formula.
Worked example
A group of 5 volunteers selects a chair and a different secretary. How many assignments of the two roles are possible? There are 5 choices for chair. Afterward 4 people remain for secretary. The roles differ, so multiply without halving: 5×4=20.
A contrasting decision
A team of two people is selected from 6 candidates. Neither member has a distinct role. How many different teams are possible? Ordered selection gives 6×5 possibilities. Each unordered pair appears twice, once in each order. Divide by two: 30/2=15 teams.
Diagnose a competing method
The first example’s tempting answer 10 follows this mistake: Count unordered pairs despite the roles being different. The same two people in exchanged roles produce different assignments.
Transfer before revealing
A cafe offers 5 main dishes and 4 sides. A meal uses one of each, but 3 specific main-side pairs are unavailable. How many meals can be ordered? Pause and identify the requested quantity. Solution: Without restrictions there are 5×4=20 pairs. Each unavailable pair removes one whole choice, and 3 distinct pairs are unavailable. The remaining count is 20−3=17.
Check the boundary of the method
Subtract that overlap once before forming the probability of either event. Decide whether swapping the two people creates a new outcome. Compare the answer with the question’s units and conditions. One correct example does not prove an identity or a statistical generalization.
A group of 5 volunteers selects a chair and a different secretary. How many assignments of the two roles are possible?
- There are 5 choices for chair.
- Afterward 4 people remain for secretary.
- The roles differ, so multiply without halving: 5×4=20.
Compare the other answer choices
Count only the chair selection.
Add sequential choice counts instead of multiplying them.
Count unordered pairs despite the roles being different.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
Show the model and intermediate calculations, then refute one competing method. This written extension is not automatically scored.
This draft stays on this page only. Use paper if you want to keep your working.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GED Mathematical Reasoning Assessment Guide for Educators