mediumConceptEntry
Permutations, combinations and probability in aptitude tests: how do you count without double counting?
Decide first whether order matters: permutations if it does, combinations if it does not. Divide out repeated items, and for "at least one" count the complement and subtract. Picking one item first and then the rest freely double counts, because each outcome is reached several ways.
4 min readUpdated 2026-10-10Target archetype: Enterprise Captive
Answer
Short answer
Ask one question first: does order matter? If it does, count permutations; if it does not, count combinations. Permutations and combinations questions then go wrong in two ways: counting the same outcome more than once, or adding probabilities that overlap. For "at least one", count the outcomes with none and subtract from the total.
Order matters: P(n, r) = n! / (n - r)!
Order does not: C(n, r) = n! / (r! (n - r)!)
At least one: total - none
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