Calculate the number of combinations (nCr) and permutations (nPr) for a set.
Combinations count how many ways to choose r items from n when order doesn't matter: nCr = n! ÷ (r! × (n−r)!). Permutations count the same selection when order does matter: nPr = n! ÷ (n−r)!.
Permutations are always equal to or larger than combinations for the same n and r, since every combination can be arranged in multiple orders — the permutation count is the combination count multiplied by r!.
The distinction comes up constantly: picking a 3-person committee from 10 people is a combination (order irrelevant); picking a 1st, 2nd, and 3rd place finisher from 10 racers is a permutation (order matters completely).
Choosing 3 items from 10: there are 120 combinations (order doesn't matter) but 720 permutations (order matters) — permutations are combinations × r! (here, × 3! = ×6).
Ask whether swapping the order of the chosen items would count as a different outcome. If yes (like race placements), it's a permutation. If no (like committee membership), it's a combination.
0! is defined as 1 by convention, which is what makes the nCr and nPr formulas work correctly when r equals n — choosing all n items from a set of n has exactly one combination.
Factorials grow extremely quickly — 10! is already over 3.6 million — which is why combination and permutation counts can become enormous even for modest values of n.