Arrangement

An arrangement is a way of choosing part of the elements of a set while taking into account the order in which they are kept. You therefore select a subgroup, but unlike a combination, the position of each chosen element matters. Two selections made up of the same elements but laid out differently count as two distinct arrangements.

The intuition is to place yourself between two known cases. A permutation orders the whole of the set, leaving nothing out. A combination chooses part of the elements while ignoring order. An arrangement takes up the idea of choosing only a part, but adds to it the taking into account of order, like a permutation applied to the subgroup kept.

To count the arrangements of k elements from n, you reason by successive positions. There are n choices for the first place, then n minus one for the second, and so on, until k positions have been filled. You therefore multiply k decreasing factors starting from n. For example, choosing a first and a second from 4 people gives 4 times 3, that is 12 possible arrangements. You obtain considerably more than the 6 corresponding combinations, precisely because here order separates cases that the combination would have grouped together.

The relationship with the other notions is clear. If you choose as many elements as there are in the set, the arrangement becomes a permutation again and is counted by a factorial. If you decide to forget order, you divide by the factorial of the number of chosen elements and the arrangement becomes a combination. An arrangement is therefore the natural bridge between these two ideas.

This tool serves to count, for example, the possible podiums of a competition, where designating a first, a second and a third place is nothing like choosing a simple trio. On a random draw site, the arrangement describes any situation where chance assigns ordered places: successively drawing a grand winner, then a finalist, then a substitute is an arrangement, because the order of the draw determines a different role for each person chosen.