Combinatorics and Probability
Core ideas
Distinguish permutations (order matters) from combinations (order does not), apply the multiplication and addition principles, and use complementary counting when direct counting is messy. In probability, know sample spaces, equally likely outcomes, independent events (P(A and B) = P(A)P(B)), mutually exclusive events (P(A or B) = P(A) + P(B)), and conditional probability. Recognize when to use inclusion-exclusion.
Traps to watch
- Using permutations when order is irrelevant (or vice versa).
- Double counting in "at least one" problems; use complements instead.
- Assuming independence without justification.
- Forgetting that "or" events may overlap, requiring subtraction of the intersection.
- Treating identical objects as distinct.
Worked example
How many ways can a committee of 3 be chosen from 8 people? Order does not matter, so use C(8,3) = 8!/(3!5!) = 56. If instead the question asked for president, vice president, and secretary, order matters: P(8,3) = 876 = 336. The wording "committee" versus "officers" signals which to use.
Rule of thumb
Ask whether swapping two chosen items produces a different outcome. If yes, use permutations; if no, use combinations. For "at least one," compute 1 minus the probability of none.