# Difference between permutation and combination?

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.

• Counts the number of ways of choosing and ordering the choices. – Arturo Magidin Jul 6 '12 at 2:44

If you see how combinations work then you're most of the way there. Say I want to pick 3 letters out of ABCDE. There are $C(5,3)$ ways of doing this. But if order matters, then several things that I counted as the same are now different. Picking $ABC$ now generates $ABC,ACB,BAC,BCA,CBA,CAB$ as different choices, when they weren't before. How many different choices are there? Well that's the number of ways I can rearrange the $r$ chosen letters, which is $r!$. So if permutations matter:

$$P(n,r)=r!\cdot C(n,r)=\frac{r!n!}{(n-r)!r!}=\frac{n!}{(n-r)!}$$

Permutations are the number of different ordered selections of $r$ elements from a set of $n$.

I just want to demonstrate via a figure the difference between permutation and combination. Hope helpful for you. Permutations assume there are no repetition of the same objects. One way to see why the formula holds is as follows:

Say you have $n$ objects and want to choose $r$ from them.

You can choose any of the $n$ for the first one,

You can choose any of the $n-1$ for the second one,

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.

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You can choose any of the $n-r+1$ for the $r^{th}$ one.

Multiplying all these altogether gives you the number of possibilities: $$n(n-1)\cdots (n-r+1)=\frac{n!}{(n-r)!}$$

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.