Combination

A combination is a way of choosing a certain number of elements from within a set without worrying about the order in which they are kept. Only the final group obtained matters: two selections that contain exactly the same elements, even if drawn in a different order, count as one and the same combination.

The intuition is simple: as soon as rearranging the chosen elements does not create a new possibility, you are dealing with a combination. The groups {1, 2, 3}, {3, 2, 1} and {2, 1, 3} are identical, because they bring together the same three numbers. This is what distinguishes a combination from an arrangement, where each order would instead form a distinct case.

To count combinations, you choose k elements from n. The idea behind the calculation is to first count all the ordered selections, then divide by the number of possible orders of the k elements kept, that is, by the factorial of k, since these orders must not be counted separately. With a small example: choosing 2 people from 4 gives 6 combinations, namely the pairs AB, AC, AD, BC, BD and CD. The pair AB is the same as BA, which explains why you do not obtain any more possibilities.

It is useful to keep in mind the distinction between the three neighbouring notions. A permutation orders the whole of a set and takes order into account. An arrangement chooses part of the elements while taking order into account. A combination, for its part, chooses part of the elements without taking order into account. Moving from an arrangement to a combination amounts precisely to erasing the order information.

Combinations are at the heart of games where you tick numbers without worrying about their order, such as the lottery. Calculating the total number of possible grids directly gives the denominator used to estimate a probability of winning. On a random draw site, this reasoning also sheds light on situations where you select a group of winners with no hierarchy among them: what matters is the list of those chosen, not the order in which chance designated them.