When only the set matters
A combination counts selections: committees, card hands, lottery tickets: anywhere {A, B, C} and {C, B, A} are the same outcome. The count starts from permutations and then divides out the redundancy: every chosen set of r items can be arranged r! ways, and all of them collapse into one combination.
Choose 3 from 10: the 720 orderings ÷ 3! = 120 combinations. Scale it up and the numbers get famous: a 5-card hand from 52 cards is C(52, 5) = 2,598,960: the denominator of every poker probability you have ever heard. Symmetry worth knowing: C(n, r) = C(n, n−r), because choosing who is in is the same act as choosing who is out.
Combinations with repetition is a different formula (stars and bars: C(n+r−1, r)): the calculator's repetition toggle switches to it. Sampling soup ladles and picking committee members repeat differently; check the model before the arithmetic.
Frequently asked questions
Permutation or combination: how do I decide?
Ask: if I swap two chosen items, does the outcome change? Podium: yes (permutation). Committee: no (combination).
Why is C(n, 0) = 1?
There is exactly one way to choose nothing. The empty selection is a selection.