What Is a Combination?
Definition
Combination: A choice of r items from n different items in which the order does not matter. Their number is C(n, r) = n! / (r! × (n − r)!); a team of 3 from 10 people can be chosen in 120 ways.
A combination is a choice of r items from n different items in which the order does not matter: {A, B} and {B, A} are the same choice. Combinations count teams, hands of cards and lottery tickets. A team of 3 from 10 people can be chosen in 120 ways; if the order mattered, there would be 720.
Where it appears in the formulas
C(n, r) = n! / (r! × (n − r)!), also written nCr. The combinations calculator uses it: C(10, 3) = 3,628,800 / (6 × 5,040) = 120. With repetition, when an item can be chosen more than once, the count is C(n + r − 1, r).
How it differs from a permutation
In a permutation the order matters, so each of the r! orders of a choice counts separately. That is why P(n, r) = C(n, r) × r!: 720 = 120 × 6.