Permutation

A permutation is a way of ordering all the elements of a set. In other words, you take all the available elements and decide in what order to arrange them. Each different arrangement constitutes a distinct permutation, because here order makes all the difference.

The intuition can be summed up in one sentence: if moving an element changes the result, you are indeed dealing with permutations. The sequences {1, 2, 3} and {3, 2, 1} contain the same numbers, but since their order differs, they count as two separate permutations. This is the opposite of a combination, where order would be ignored and these two sequences would form only one.

To count the permutations of n elements, you reason in successive steps. There are n choices for the first position, then n minus one choices for the second since one element is already placed, then n minus two for the third, and so on until exhaustion. Multiplying these choices amounts to calculating the factorial of n, written n!. For example, ordering 3 players gives 3! equals 6 possible orders, and shuffling 4 tokens gives 4! equals 24 different arrangements.

Three neighbouring notions must be distinguished. A permutation orders all the elements of the set. An arrangement chooses only part of them, but always taking order into account. A combination also chooses part of the elements, but this time without taking order into account. A permutation is therefore the special case where you arrange the complete set.

Permutations are used to count the possible orders, which corresponds directly to several random draw tools. Determining the running order of a list of participants, setting a playing order, or randomly shuffling a list before displaying it all amount to choosing a permutation from among all those that the factorial allows you to count. Understanding this counting helps to gauge just how many equally plausible results a shuffle can produce.