What does an "arrangement" mean in combinatorics?

An arrangement in combinatorics is a way to select and order elements from a set.

#Formal definition

An arrangement of n taken k at a time is an ordered selection of k distinct elements taken from n distinct elements.

Key words:

• elements are not repeated;
• order matters.

#Example

Suppose there are 3 elements: {A, B, C}. Arrangements of 2:

• AB
• BA
• AC
• CA
• BC
• CB

Total:

A32=3!(3−2)!=6A_3^2 = \frac{3!}{(3 - 2)!} = 6A32​=(3−2)!3!​=6

#Difference from other concepts

#In the context of algorithms

Arrangements come up in tasks such as:

• generating all variants of length k without repetition;
• enumerating states while taking order into account;
• brute-force solutions with a fixed sequence length.

Short version for an interview:

An arrangement is an ordered selection of k distinct elements out of n: order matters, no repetitions.