TITLE: Difference between permutation and combination?
QUESTION [8 upvotes]: Permutation:
$$P(n,r) = \frac{n!}{(n-r)!}$$
Combination:
$$C(n,r) = \frac{n!}{(n-r)!r!}$$
Apparently, you use combination when the order doesn't matter. Great. I see how a combination will give you all the possible well, combinations. However, I don't see what exactly does a permutation do then.

REPLY [2 votes]: There is some common confusion between these two terms.
The word "permutation" in general refers to one of three things depending on context. It can mean the order (arrangement) of a set as in combinatorics. Or it can refer to an arrangement of a subset of a given size as also in combinatorics. Or it can refer to an OPERATION of REarrangement in a space of 
such operations as in group theory, which is a very different thing.
"Combination" refers only to the makeup (constituency) of a subset with no concept of order. 
The term "combinations" refers to the number of subsets of a given size containing different constituents.
The combinatorial meanings of both of these terms are still valid when applied to sets with duplicate elements. A permutation (arrangement or rearrangement) can apply to a set or subset that contains duplicates. But "combination" usually assumes distinct elements in the subset, though the original set can contain duplicates. Of course, if so indicated, a combination could also contain duplicates. The presence of duplicates affects the combinatorial formulas for all of these.