Permutations and combinations

Permutations and combinations answer the question "how many ways can this happen?" when choosing items from a larger set. The difference is whether order matters. A permutation counts ordered arrangements — gold, silver, and bronze medalists are a permutation, because finishing first differs from finishing third. A combination counts unordered selections — a committee of three people is a combination, because the same three people form the same committee regardless of the order they were picked.

Both formulas build on factorials. The number of permutations of r items chosen from n is nPr = n! / (n − r)!, and the number of combinations is nCr = n! / (r!(n − r)!). The extra r! in the combination formula divides out the duplicate orderings of each selection. For example, choosing 3 medalists from 8 runners gives 8P3 = 8 × 7 × 6 = 336 arrangements, while choosing a 3-person committee from 8 people gives 8C3 = 336 / 3! = 56 groups.

For any given n and r, there are always at least as many permutations as combinations, since each unordered group can be arranged in multiple orders. A quick test for word problems: if swapping two selected items produces a genuinely different outcome (seating orders, passwords, rankings), use permutations; if not (committees, pizza toppings, hands of cards), use combinations.

Counting problems appear regularly on the GRE and ACT and are a cornerstone of AMC 8 combinatorics. Exam questions typically hinge on recognizing which formula applies, so practice classifying problems as order-matters or order-doesn't before computing.