Permutations and Combinations

Permutations count ordered arrangements and combinations count unordered selections under specified repetition and distinguishability rules. The multiplication principle counts independent choice stages when all combinations are allowed.

Counting begins by defining what makes two outcomes different. Choosing r distinct objects from n without order gives n!/[r!(n − r)!] combinations; assigning those objects to ordered positions gives n!/(n − r)! permutations. Allowing repetition changes the calculation, and constraints can remove many otherwise possible outcomes.

Counts become probabilities only when the sampling mechanism is specified. Dividing favorable cases by total cases requires equally likely elementary outcomes or appropriate weights. A large design or password space therefore does not by itself establish practical difficulty, useful diversity, or security; feasibility and nonuniform selection can change the relevant space.

When to use it

When analyzing data; when building predictive models; when evaluating statistical claims; when quantifying uncertainty.

How it can help

Define what makes outcomes distinct, whether order and repetition matter, and which constraints apply. Check a small case before using the formula.