Combinatorics, probability theory and statistics
Counting, then measuring chance
Multiplication principle: if a choice splits into independent stages with m and n options, the total is mn. Permutations count ordered arrangements: P(n, k) = n!/(n − k)!, and arranging all n objects in a row gives n!. Combinations count unordered selections: C(n, k) = n!/(k!(n − k)!). Decide first whether order matters and whether repetition is allowed - that decision selects the formula.
Probability: with equally likely outcomes, P(A) = favourable/total. Use the complement for "at least one": P(at least one) = 1 − P(none). For mutually exclusive events P(A∪B) = P(A) + P(B); for independent events P(A∩B) = P(A)P(B). Without replacement, the denominator shrinks at each draw.
Statistics: the mean is sensitive to outliers, the median is not; always sort the data before finding the median. The mode is the most frequent value; the range is max − min.
Traps. Using permutations where order is irrelevant, or the reverse. Adding probabilities of events that can occur together. Forgetting to reduce the sample space after a draw. Reporting the median of unsorted data.
Worked example. Two dice: P(at least one six) = 1 − P(no six) = 1 − (5/6)² = 11/36, not 1/6 + 1/6 = 1/3.
Rule of thumb: "at least one" almost always means "use the complement".