A reliable method
Decide whether order matters before counting. Use combinations for unordered selections and permutations for distinct positions. For a condition, rebuild the denominator from the restricted group.
The common trap
Independent events and mutually exclusive events are different. Adding probabilities double-counts overlap unless the overlap is subtracted.
Make the connection
Write a short tree or table for a small case. If a probability is outside zero to one, revisit the count or the conditioning event.
Contrast the cases
From five people, choosing a two-person committee gives 10 unordered pairs, but filling president and secretary gives 20 ordered assignments. For two red and one blue token, P(two red without replacement)=(2/3)(1/2)=1/3; replacing the first token would instead give 4/9. “Given the first draw is red” also changes the denominator for the second draw. Count the permitted sample space before multiplying.
Transfer and explain
Transfer: among four red and two blue tokens, P(two red without replacement)= (4/6)(3/5)=2/5. State why the second numerator and denominator both decrease.
A club has 7 members. In how many ways can it choose a president and a different secretary?
- There are 7 choices for president.
- After that choice, 6 members remain eligible for secretary.
- Multiply: 7 × 6 = 42.
Compare the other answer choices
Counts only the president choice.
Counts all n people separately for each role, ignores the distinct-person restriction, then adds n+n instead of multiplying eligible choices.
Allows one person to hold both distinct roles.
Arranges every member, not only the two roles.
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.
GRE Quantitative Reasoning overviewSiyavula Grade 10 mathematics foundation reference