"Story" proof for $\frac{(2n)!}{2^nn!}=(2n-1)(2n-3)\cdots3\cdot1$ I intuitively understand how this works if the story is that you're counting each way to pair each member of a group of size $(2n)!$. But, I am having trouble walking through the computation and the rationale for each step, especially for the $2^n$ term. 
Example: $n=2$
For example, if the group consisted of $4$ people, $n=2$, and the solution is $3$. This makes sense from the following, step-wise calculations:


*

*Counting the permutations of groups of size $n$ taken from $2n$ is $\frac{(2n!)}{n!}=\frac{(2\cdot2)!)}{2!}=12$.

*We divide this by $2^1$ because we "assign" half of the permutations to the other half. Think: each "team" is a permutation, and we need each "team" to play another "team" (i.e., a permutation that does not have the same elements of the former permutation which it is being paired with). (E.g., AB "paired up" with CD). These pairings are called "games". Therefore, $\frac{12}{2}=6$.

*So, now we see that we have counted all the "games" and we have to get rid of duplicate "games." We divide by $2^1$ again because each way to pair constituent members of a "game" is counted twice. If you're keeping score at home, we've explained the $2^n$ term for this example. See below for the duplicates:



AB CD is the same as BA DC.
   BC AD is the same as CB DA.
  BD CA is the same as DB AC.

We've arrived at the answer: $3=(2(2)-1)(2(2)-3)=3\cdot1$. So far, so good. Let's try another easy example.
Example: $n=3$ ... AKA where I get stuck.


*

*Same as above.

*All good here. I've got it down to $\frac{((2\cdot3)!)}{2^1\cdot3!}=60$. I know the answer is $15$.

*This is where the wheels come off for me. I've got the "games", but I have no idea why we are dividing by $2^2$. It has occurred to me that the second time I divided by $2$ in the example above (where $n=2$), it was $2!=2^1$ which eliminates all duplicate "games". This time, therefore, I keep wanting to divide by $6!$ to eliminate duplicate "games," but I know this is wrong and not the "story," per se (i.e., the "story" being how many ways to pair people in a group of $(2n)!$)


Hopefully, this makes sense and someone might be able to shed some light on why there is redundancy in counting of games (i.e., each unique "game" is counted $2^2=4$ times). How would I show this was some of the "games" being made explicit/manual counting (as I did in the example above)?
Thanks, all, in advance.
 A: We divide by $n!$ because the order of the $n$ groups is irrelevant. We divide by $2^n$ because for each of the $n$ pairs, the order of the people in the pair is also irrelevant.
A: Let $S$ be a set with $2n$ elements, so that $(2n)!$ is the number of permutations of $S$.
Dividing by $2^n$ produces $\frac{(2n)!}{2^n}$, the number of ordered partitions of $S$ into $n$ pairs.
Then dividing by $n!$ produces $\frac{(2n)!}{2^n\cdot n!}$, the number of unordered partitions of $S$ into $n$ pairs.

For $n=3$, let $S=\{\textsf{A,B,C,D,E,F}\}$. Then the $720$ permutations of $S$ are

(A, B, C, D, E, F)
(A, B, C, D, F, E)
(A, B, C, E, D, F)
(A, B, C, E, F, D)
(A, B, C, F, D, E)
(A, B, C, F, E, D)
(A, B, D, C, E, F)
(A, B, D, C, F, E)
(A, B, D, E, C, F)
(A, B, D, E, F, C)
(A, B, D, F, C, E)
(A, B, D, F, E, C)
(A, B, E, C, D, F)
(A, B, E, C, F, D)
(A, B, E, D, C, F)
(A, B, E, D, F, C)
(A, B, E, F, C, D)
(A, B, E, F, D, C)
(A, B, F, C, D, E)
(A, B, F, C, E, D)
(A, B, F, D, C, E)
(A, B, F, D, E, C)
(A, B, F, E, C, D)
(A, B, F, E, D, C)
(A, C, B, D, E, F)
(A, C, B, D, F, E)
(A, C, B, E, D, F)
(A, C, B, E, F, D)
(A, C, B, F, D, E)
(A, C, B, F, E, D)
(A, C, D, B, E, F)
(A, C, D, B, F, E)
(A, C, D, E, B, F)
(A, C, D, E, F, B)
(A, C, D, F, B, E)
(A, C, D, F, E, B)
(A, C, E, B, D, F)
(A, C, E, B, F, D)
(A, C, E, D, B, F)
(A, C, E, D, F, B)
(A, C, E, F, B, D)
(A, C, E, F, D, B)
(A, C, F, B, D, E)
(A, C, F, B, E, D)
(A, C, F, D, B, E)
(A, C, F, D, E, B)
(A, C, F, E, B, D)
(A, C, F, E, D, B)
(A, D, B, C, E, F)
(A, D, B, C, F, E)
(A, D, B, E, C, F)
(A, D, B, E, F, C)
(A, D, B, F, C, E)
(A, D, B, F, E, C)
(A, D, C, B, E, F)
(A, D, C, B, F, E)
(A, D, C, E, B, F)
(A, D, C, E, F, B)
(A, D, C, F, B, E)
(A, D, C, F, E, B)
(A, D, E, B, C, F)
(A, D, E, B, F, C)
(A, D, E, C, B, F)
(A, D, E, C, F, B)
(A, D, E, F, B, C)
(A, D, E, F, C, B)
(A, D, F, B, C, E)
(A, D, F, B, E, C)
(A, D, F, C, B, E)
(A, D, F, C, E, B)
(A, D, F, E, B, C)
(A, D, F, E, C, B)
(A, E, B, C, D, F)
(A, E, B, C, F, D)
(A, E, B, D, C, F)
(A, E, B, D, F, C)
(A, E, B, F, C, D)
(A, E, B, F, D, C)
(A, E, C, B, D, F)
(A, E, C, B, F, D)
(A, E, C, D, B, F)
(A, E, C, D, F, B)
(A, E, C, F, B, D)
(A, E, C, F, D, B)
(A, E, D, B, C, F)
(A, E, D, B, F, C)
(A, E, D, C, B, F)
(A, E, D, C, F, B)
(A, E, D, F, B, C)
(A, E, D, F, C, B)
(A, E, F, B, C, D)
(A, E, F, B, D, C)
(A, E, F, C, B, D)
(A, E, F, C, D, B)
(A, E, F, D, B, C)
(A, E, F, D, C, B)
(A, F, B, C, D, E)
(A, F, B, C, E, D)
(A, F, B, D, C, E)
(A, F, B, D, E, C)
(A, F, B, E, C, D)
(A, F, B, E, D, C)
(A, F, C, B, D, E)
(A, F, C, B, E, D)
(A, F, C, D, B, E)
(A, F, C, D, E, B)
(A, F, C, E, B, D)
(A, F, C, E, D, B)
(A, F, D, B, C, E)
(A, F, D, B, E, C)
(A, F, D, C, B, E)
(A, F, D, C, E, B)
(A, F, D, E, B, C)
(A, F, D, E, C, B)
(A, F, E, B, C, D)
(A, F, E, B, D, C)
(A, F, E, C, B, D)
(A, F, E, C, D, B)
(A, F, E, D, B, C)
(A, F, E, D, C, B)
(B, A, C, D, E, F)
(B, A, C, D, F, E)
(B, A, C, E, D, F)
(B, A, C, E, F, D)
(B, A, C, F, D, E)
(B, A, C, F, E, D)
(B, A, D, C, E, F)
(B, A, D, C, F, E)
(B, A, D, E, C, F)
(B, A, D, E, F, C)
(B, A, D, F, C, E)
(B, A, D, F, E, C)
(B, A, E, C, D, F)
(B, A, E, C, F, D)
(B, A, E, D, C, F)
(B, A, E, D, F, C)
(B, A, E, F, C, D)
(B, A, E, F, D, C)
(B, A, F, C, D, E)
(B, A, F, C, E, D)
(B, A, F, D, C, E)
(B, A, F, D, E, C)
(B, A, F, E, C, D)
(B, A, F, E, D, C)
(B, C, A, D, E, F)
(B, C, A, D, F, E)
(B, C, A, E, D, F)
(B, C, A, E, F, D)
(B, C, A, F, D, E)
(B, C, A, F, E, D)
(B, C, D, A, E, F)
(B, C, D, A, F, E)
(B, C, D, E, A, F)
(B, C, D, E, F, A)
(B, C, D, F, A, E)
(B, C, D, F, E, A)
(B, C, E, A, D, F)
(B, C, E, A, F, D)
(B, C, E, D, A, F)
(B, C, E, D, F, A)
(B, C, E, F, A, D)
(B, C, E, F, D, A)
(B, C, F, A, D, E)
(B, C, F, A, E, D)
(B, C, F, D, A, E)
(B, C, F, D, E, A)
(B, C, F, E, A, D)
(B, C, F, E, D, A)
(B, D, A, C, E, F)
(B, D, A, C, F, E)
(B, D, A, E, C, F)
(B, D, A, E, F, C)
(B, D, A, F, C, E)
(B, D, A, F, E, C)
(B, D, C, A, E, F)
(B, D, C, A, F, E)
(B, D, C, E, A, F)
(B, D, C, E, F, A)
(B, D, C, F, A, E)
(B, D, C, F, E, A)
(B, D, E, A, C, F)
(B, D, E, A, F, C)
(B, D, E, C, A, F)
(B, D, E, C, F, A)
(B, D, E, F, A, C)
(B, D, E, F, C, A)
(B, D, F, A, C, E)
(B, D, F, A, E, C)
(B, D, F, C, A, E)
(B, D, F, C, E, A)
(B, D, F, E, A, C)
(B, D, F, E, C, A)
(B, E, A, C, D, F)
(B, E, A, C, F, D)
(B, E, A, D, C, F)
(B, E, A, D, F, C)
(B, E, A, F, C, D)
(B, E, A, F, D, C)
(B, E, C, A, D, F)
(B, E, C, A, F, D)
(B, E, C, D, A, F)
(B, E, C, D, F, A)
(B, E, C, F, A, D)
(B, E, C, F, D, A)
(B, E, D, A, C, F)
(B, E, D, A, F, C)
(B, E, D, C, A, F)
(B, E, D, C, F, A)
(B, E, D, F, A, C)
(B, E, D, F, C, A)
(B, E, F, A, C, D)
(B, E, F, A, D, C)
(B, E, F, C, A, D)
(B, E, F, C, D, A)
(B, E, F, D, A, C)
(B, E, F, D, C, A)
(B, F, A, C, D, E)
(B, F, A, C, E, D)
(B, F, A, D, C, E)
(B, F, A, D, E, C)
(B, F, A, E, C, D)
(B, F, A, E, D, C)
(B, F, C, A, D, E)
(B, F, C, A, E, D)
(B, F, C, D, A, E)
(B, F, C, D, E, A)
(B, F, C, E, A, D)
(B, F, C, E, D, A)
(B, F, D, A, C, E)
(B, F, D, A, E, C)
(B, F, D, C, A, E)
(B, F, D, C, E, A)
(B, F, D, E, A, C)
(B, F, D, E, C, A)
(B, F, E, A, C, D)
(B, F, E, A, D, C)
(B, F, E, C, A, D)
(B, F, E, C, D, A)
(B, F, E, D, A, C)
(B, F, E, D, C, A)
(C, A, B, D, E, F)
(C, A, B, D, F, E)
(C, A, B, E, D, F)
(C, A, B, E, F, D)
(C, A, B, F, D, E)
(C, A, B, F, E, D)
(C, A, D, B, E, F)
(C, A, D, B, F, E)
(C, A, D, E, B, F)
(C, A, D, E, F, B)
(C, A, D, F, B, E)
(C, A, D, F, E, B)
(C, A, E, B, D, F)
(C, A, E, B, F, D)
(C, A, E, D, B, F)
(C, A, E, D, F, B)
(C, A, E, F, B, D)
(C, A, E, F, D, B)
(C, A, F, B, D, E)
(C, A, F, B, E, D)
(C, A, F, D, B, E)
(C, A, F, D, E, B)
(C, A, F, E, B, D)
(C, A, F, E, D, B)
(C, B, A, D, E, F)
(C, B, A, D, F, E)
(C, B, A, E, D, F)
(C, B, A, E, F, D)
(C, B, A, F, D, E)
(C, B, A, F, E, D)
(C, B, D, A, E, F)
(C, B, D, A, F, E)
(C, B, D, E, A, F)
(C, B, D, E, F, A)
(C, B, D, F, A, E)
(C, B, D, F, E, A)
(C, B, E, A, D, F)
(C, B, E, A, F, D)
(C, B, E, D, A, F)
(C, B, E, D, F, A)
(C, B, E, F, A, D)
(C, B, E, F, D, A)
(C, B, F, A, D, E)
(C, B, F, A, E, D)
(C, B, F, D, A, E)
(C, B, F, D, E, A)
(C, B, F, E, A, D)
(C, B, F, E, D, A)
(C, D, A, B, E, F)
(C, D, A, B, F, E)
(C, D, A, E, B, F)
(C, D, A, E, F, B)
(C, D, A, F, B, E)
(C, D, A, F, E, B)
(C, D, B, A, E, F)
(C, D, B, A, F, E)
(C, D, B, E, A, F)
(C, D, B, E, F, A)
(C, D, B, F, A, E)
(C, D, B, F, E, A)
(C, D, E, A, B, F)
(C, D, E, A, F, B)
(C, D, E, B, A, F)
(C, D, E, B, F, A)
(C, D, E, F, A, B)
(C, D, E, F, B, A)
(C, D, F, A, B, E)
(C, D, F, A, E, B)
(C, D, F, B, A, E)
(C, D, F, B, E, A)
(C, D, F, E, A, B)
(C, D, F, E, B, A)
(C, E, A, B, D, F)
(C, E, A, B, F, D)
(C, E, A, D, B, F)
(C, E, A, D, F, B)
(C, E, A, F, B, D)
(C, E, A, F, D, B)
(C, E, B, A, D, F)
(C, E, B, A, F, D)
(C, E, B, D, A, F)
(C, E, B, D, F, A)
(C, E, B, F, A, D)
(C, E, B, F, D, A)
(C, E, D, A, B, F)
(C, E, D, A, F, B)
(C, E, D, B, A, F)
(C, E, D, B, F, A)
(C, E, D, F, A, B)
(C, E, D, F, B, A)
(C, E, F, A, B, D)
(C, E, F, A, D, B)
(C, E, F, B, A, D)
(C, E, F, B, D, A)
(C, E, F, D, A, B)
(C, E, F, D, B, A)
(C, F, A, B, D, E)
(C, F, A, B, E, D)
(C, F, A, D, B, E)
(C, F, A, D, E, B)
(C, F, A, E, B, D)
(C, F, A, E, D, B)
(C, F, B, A, D, E)
(C, F, B, A, E, D)
(C, F, B, D, A, E)
(C, F, B, D, E, A)
(C, F, B, E, A, D)
(C, F, B, E, D, A)
(C, F, D, A, B, E)
(C, F, D, A, E, B)
(C, F, D, B, A, E)
(C, F, D, B, E, A)
(C, F, D, E, A, B)
(C, F, D, E, B, A)
(C, F, E, A, B, D)
(C, F, E, A, D, B)
(C, F, E, B, A, D)
(C, F, E, B, D, A)
(C, F, E, D, A, B)
(C, F, E, D, B, A)
(D, A, B, C, E, F)
(D, A, B, C, F, E)
(D, A, B, E, C, F)
(D, A, B, E, F, C)
(D, A, B, F, C, E)
(D, A, B, F, E, C)
(D, A, C, B, E, F)
(D, A, C, B, F, E)
(D, A, C, E, B, F)
(D, A, C, E, F, B)
(D, A, C, F, B, E)
(D, A, C, F, E, B)
(D, A, E, B, C, F)
(D, A, E, B, F, C)
(D, A, E, C, B, F)
(D, A, E, C, F, B)
(D, A, E, F, B, C)
(D, A, E, F, C, B)
(D, A, F, B, C, E)
(D, A, F, B, E, C)
(D, A, F, C, B, E)
(D, A, F, C, E, B)
(D, A, F, E, B, C)
(D, A, F, E, C, B)
(D, B, A, C, E, F)
(D, B, A, C, F, E)
(D, B, A, E, C, F)
(D, B, A, E, F, C)
(D, B, A, F, C, E)
(D, B, A, F, E, C)
(D, B, C, A, E, F)
(D, B, C, A, F, E)
(D, B, C, E, A, F)
(D, B, C, E, F, A)
(D, B, C, F, A, E)
(D, B, C, F, E, A)
(D, B, E, A, C, F)
(D, B, E, A, F, C)
(D, B, E, C, A, F)
(D, B, E, C, F, A)
(D, B, E, F, A, C)
(D, B, E, F, C, A)
(D, B, F, A, C, E)
(D, B, F, A, E, C)
(D, B, F, C, A, E)
(D, B, F, C, E, A)
(D, B, F, E, A, C)
(D, B, F, E, C, A)
(D, C, A, B, E, F)
(D, C, A, B, F, E)
(D, C, A, E, B, F)
(D, C, A, E, F, B)
(D, C, A, F, B, E)
(D, C, A, F, E, B)
(D, C, B, A, E, F)
(D, C, B, A, F, E)
(D, C, B, E, A, F)
(D, C, B, E, F, A)
(D, C, B, F, A, E)
(D, C, B, F, E, A)
(D, C, E, A, B, F)
(D, C, E, A, F, B)
(D, C, E, B, A, F)
(D, C, E, B, F, A)
(D, C, E, F, A, B)
(D, C, E, F, B, A)
(D, C, F, A, B, E)
(D, C, F, A, E, B)
(D, C, F, B, A, E)
(D, C, F, B, E, A)
(D, C, F, E, A, B)
(D, C, F, E, B, A)
(D, E, A, B, C, F)
(D, E, A, B, F, C)
(D, E, A, C, B, F)
(D, E, A, C, F, B)
(D, E, A, F, B, C)
(D, E, A, F, C, B)
(D, E, B, A, C, F)
(D, E, B, A, F, C)
(D, E, B, C, A, F)
(D, E, B, C, F, A)
(D, E, B, F, A, C)
(D, E, B, F, C, A)
(D, E, C, A, B, F)
(D, E, C, A, F, B)
(D, E, C, B, A, F)
(D, E, C, B, F, A)
(D, E, C, F, A, B)
(D, E, C, F, B, A)
(D, E, F, A, B, C)
(D, E, F, A, C, B)
(D, E, F, B, A, C)
(D, E, F, B, C, A)
(D, E, F, C, A, B)
(D, E, F, C, B, A)
(D, F, A, B, C, E)
(D, F, A, B, E, C)
(D, F, A, C, B, E)
(D, F, A, C, E, B)
(D, F, A, E, B, C)
(D, F, A, E, C, B)
(D, F, B, A, C, E)
(D, F, B, A, E, C)
(D, F, B, C, A, E)
(D, F, B, C, E, A)
(D, F, B, E, A, C)
(D, F, B, E, C, A)
(D, F, C, A, B, E)
(D, F, C, A, E, B)
(D, F, C, B, A, E)
(D, F, C, B, E, A)
(D, F, C, E, A, B)
(D, F, C, E, B, A)
(D, F, E, A, B, C)
(D, F, E, A, C, B)
(D, F, E, B, A, C)
(D, F, E, B, C, A)
(D, F, E, C, A, B)
(D, F, E, C, B, A)
(E, A, B, C, D, F)
(E, A, B, C, F, D)
(E, A, B, D, C, F)
(E, A, B, D, F, C)
(E, A, B, F, C, D)
(E, A, B, F, D, C)
(E, A, C, B, D, F)
(E, A, C, B, F, D)
(E, A, C, D, B, F)
(E, A, C, D, F, B)
(E, A, C, F, B, D)
(E, A, C, F, D, B)
(E, A, D, B, C, F)
(E, A, D, B, F, C)
(E, A, D, C, B, F)
(E, A, D, C, F, B)
(E, A, D, F, B, C)
(E, A, D, F, C, B)
(E, A, F, B, C, D)
(E, A, F, B, D, C)
(E, A, F, C, B, D)
(E, A, F, C, D, B)
(E, A, F, D, B, C)
(E, A, F, D, C, B)
(E, B, A, C, D, F)
(E, B, A, C, F, D)
(E, B, A, D, C, F)
(E, B, A, D, F, C)
(E, B, A, F, C, D)
(E, B, A, F, D, C)
(E, B, C, A, D, F)
(E, B, C, A, F, D)
(E, B, C, D, A, F)
(E, B, C, D, F, A)
(E, B, C, F, A, D)
(E, B, C, F, D, A)
(E, B, D, A, C, F)
(E, B, D, A, F, C)
(E, B, D, C, A, F)
(E, B, D, C, F, A)
(E, B, D, F, A, C)
(E, B, D, F, C, A)
(E, B, F, A, C, D)
(E, B, F, A, D, C)
(E, B, F, C, A, D)
(E, B, F, C, D, A)
(E, B, F, D, A, C)
(E, B, F, D, C, A)
(E, C, A, B, D, F)
(E, C, A, B, F, D)
(E, C, A, D, B, F)
(E, C, A, D, F, B)
(E, C, A, F, B, D)
(E, C, A, F, D, B)
(E, C, B, A, D, F)
(E, C, B, A, F, D)
(E, C, B, D, A, F)
(E, C, B, D, F, A)
(E, C, B, F, A, D)
(E, C, B, F, D, A)
(E, C, D, A, B, F)
(E, C, D, A, F, B)
(E, C, D, B, A, F)
(E, C, D, B, F, A)
(E, C, D, F, A, B)
(E, C, D, F, B, A)
(E, C, F, A, B, D)
(E, C, F, A, D, B)
(E, C, F, B, A, D)
(E, C, F, B, D, A)
(E, C, F, D, A, B)
(E, C, F, D, B, A)
(E, D, A, B, C, F)
(E, D, A, B, F, C)
(E, D, A, C, B, F)
(E, D, A, C, F, B)
(E, D, A, F, B, C)
(E, D, A, F, C, B)
(E, D, B, A, C, F)
(E, D, B, A, F, C)
(E, D, B, C, A, F)
(E, D, B, C, F, A)
(E, D, B, F, A, C)
(E, D, B, F, C, A)
(E, D, C, A, B, F)
(E, D, C, A, F, B)
(E, D, C, B, A, F)
(E, D, C, B, F, A)
(E, D, C, F, A, B)
(E, D, C, F, B, A)
(E, D, F, A, B, C)
(E, D, F, A, C, B)
(E, D, F, B, A, C)
(E, D, F, B, C, A)
(E, D, F, C, A, B)
(E, D, F, C, B, A)
(E, F, A, B, C, D)
(E, F, A, B, D, C)
(E, F, A, C, B, D)
(E, F, A, C, D, B)
(E, F, A, D, B, C)
(E, F, A, D, C, B)
(E, F, B, A, C, D)
(E, F, B, A, D, C)
(E, F, B, C, A, D)
(E, F, B, C, D, A)
(E, F, B, D, A, C)
(E, F, B, D, C, A)
(E, F, C, A, B, D)
(E, F, C, A, D, B)
(E, F, C, B, A, D)
(E, F, C, B, D, A)
(E, F, C, D, A, B)
(E, F, C, D, B, A)
(E, F, D, A, B, C)
(E, F, D, A, C, B)
(E, F, D, B, A, C)
(E, F, D, B, C, A)
(E, F, D, C, A, B)
(E, F, D, C, B, A)
(F, A, B, C, D, E)
(F, A, B, C, E, D)
(F, A, B, D, C, E)
(F, A, B, D, E, C)
(F, A, B, E, C, D)
(F, A, B, E, D, C)
(F, A, C, B, D, E)
(F, A, C, B, E, D)
(F, A, C, D, B, E)
(F, A, C, D, E, B)
(F, A, C, E, B, D)
(F, A, C, E, D, B)
(F, A, D, B, C, E)
(F, A, D, B, E, C)
(F, A, D, C, B, E)
(F, A, D, C, E, B)
(F, A, D, E, B, C)
(F, A, D, E, C, B)
(F, A, E, B, C, D)
(F, A, E, B, D, C)
(F, A, E, C, B, D)
(F, A, E, C, D, B)
(F, A, E, D, B, C)
(F, A, E, D, C, B)
(F, B, A, C, D, E)
(F, B, A, C, E, D)
(F, B, A, D, C, E)
(F, B, A, D, E, C)
(F, B, A, E, C, D)
(F, B, A, E, D, C)
(F, B, C, A, D, E)
(F, B, C, A, E, D)
(F, B, C, D, A, E)
(F, B, C, D, E, A)
(F, B, C, E, A, D)
(F, B, C, E, D, A)
(F, B, D, A, C, E)
(F, B, D, A, E, C)
(F, B, D, C, A, E)
(F, B, D, C, E, A)
(F, B, D, E, A, C)
(F, B, D, E, C, A)
(F, B, E, A, C, D)
(F, B, E, A, D, C)
(F, B, E, C, A, D)
(F, B, E, C, D, A)
(F, B, E, D, A, C)
(F, B, E, D, C, A)
(F, C, A, B, D, E)
(F, C, A, B, E, D)
(F, C, A, D, B, E)
(F, C, A, D, E, B)
(F, C, A, E, B, D)
(F, C, A, E, D, B)
(F, C, B, A, D, E)
(F, C, B, A, E, D)
(F, C, B, D, A, E)
(F, C, B, D, E, A)
(F, C, B, E, A, D)
(F, C, B, E, D, A)
(F, C, D, A, B, E)
(F, C, D, A, E, B)
(F, C, D, B, A, E)
(F, C, D, B, E, A)
(F, C, D, E, A, B)
(F, C, D, E, B, A)
(F, C, E, A, B, D)
(F, C, E, A, D, B)
(F, C, E, B, A, D)
(F, C, E, B, D, A)
(F, C, E, D, A, B)
(F, C, E, D, B, A)
(F, D, A, B, C, E)
(F, D, A, B, E, C)
(F, D, A, C, B, E)
(F, D, A, C, E, B)
(F, D, A, E, B, C)
(F, D, A, E, C, B)
(F, D, B, A, C, E)
(F, D, B, A, E, C)
(F, D, B, C, A, E)
(F, D, B, C, E, A)
(F, D, B, E, A, C)
(F, D, B, E, C, A)
(F, D, C, A, B, E)
(F, D, C, A, E, B)
(F, D, C, B, A, E)
(F, D, C, B, E, A)
(F, D, C, E, A, B)
(F, D, C, E, B, A)
(F, D, E, A, B, C)
(F, D, E, A, C, B)
(F, D, E, B, A, C)
(F, D, E, B, C, A)
(F, D, E, C, A, B)
(F, D, E, C, B, A)
(F, E, A, B, C, D)
(F, E, A, B, D, C)
(F, E, A, C, B, D)
(F, E, A, C, D, B)
(F, E, A, D, B, C)
(F, E, A, D, C, B)
(F, E, B, A, C, D)
(F, E, B, A, D, C)
(F, E, B, C, A, D)
(F, E, B, C, D, A)
(F, E, B, D, A, C)
(F, E, B, D, C, A)
(F, E, C, A, B, D)
(F, E, C, A, D, B)
(F, E, C, B, A, D)
(F, E, C, B, D, A)
(F, E, C, D, A, B)
(F, E, C, D, B, A)
(F, E, D, A, B, C)
(F, E, D, A, C, B)
(F, E, D, B, A, C)
(F, E, D, B, C, A)
(F, E, D, C, A, B)
(F, E, D, C, B, A)

The $90$ ordered partitions of $S$ into $3$ pairs are

({A,B}, {C,D}, {E,F})
({A,B}, {C,E}, {D,F})
({A,B}, {C,F}, {E,D})
({A,B}, {E,D}, {C,F})
({A,B}, {D,F}, {C,E})
({A,B}, {E,F}, {C,D})
({A,C}, {B,D}, {E,F})
({A,C}, {B,E}, {D,F})
({A,C}, {B,F}, {E,D})
({A,C}, {E,D}, {B,F})
({A,C}, {D,F}, {B,E})
({A,C}, {E,F}, {B,D})
({A,D}, {C,B}, {E,F})
({A,D}, {B,E}, {C,F})
({A,D}, {B,F}, {C,E})
({A,D}, {C,E}, {B,F})
({A,D}, {C,F}, {B,E})
({A,D}, {E,F}, {C,B})
({A,E}, {C,B}, {D,F})
({A,E}, {B,D}, {C,F})
({A,E}, {B,F}, {C,D})
({A,E}, {C,D}, {B,F})
({A,E}, {C,F}, {B,D})
({A,E}, {D,F}, {C,B})
({A,F}, {C,B}, {E,D})
({A,F}, {B,D}, {C,E})
({A,F}, {B,E}, {C,D})
({A,F}, {C,D}, {B,E})
({A,F}, {C,E}, {B,D})
({A,F}, {E,D}, {C,B})
({C,B}, {A,D}, {E,F})
({C,B}, {A,E}, {D,F})
({C,B}, {A,F}, {E,D})
({C,B}, {E,D}, {A,F})
({C,B}, {D,F}, {A,E})
({C,B}, {E,F}, {A,D})
({B,D}, {A,C}, {E,F})
({B,D}, {A,E}, {C,F})
({B,D}, {A,F}, {C,E})
({B,D}, {C,E}, {A,F})
({B,D}, {C,F}, {A,E})
({B,D}, {E,F}, {A,C})
({B,E}, {A,C}, {D,F})
({B,E}, {A,D}, {C,F})
({B,E}, {A,F}, {C,D})
({B,E}, {C,D}, {A,F})
({B,E}, {C,F}, {A,D})
({B,E}, {D,F}, {A,C})
({B,F}, {A,C}, {E,D})
({B,F}, {A,D}, {C,E})
({B,F}, {A,E}, {C,D})
({B,F}, {C,D}, {A,E})
({B,F}, {C,E}, {A,D})
({B,F}, {E,D}, {A,C})
({C,D}, {A,B}, {E,F})
({C,D}, {A,E}, {B,F})
({C,D}, {A,F}, {B,E})
({C,D}, {B,E}, {A,F})
({C,D}, {B,F}, {A,E})
({C,D}, {E,F}, {A,B})
({C,E}, {A,B}, {D,F})
({C,E}, {A,D}, {B,F})
({C,E}, {A,F}, {B,D})
({C,E}, {B,D}, {A,F})
({C,E}, {B,F}, {A,D})
({C,E}, {D,F}, {A,B})
({C,F}, {A,B}, {E,D})
({C,F}, {A,D}, {B,E})
({C,F}, {A,E}, {B,D})
({C,F}, {B,D}, {A,E})
({C,F}, {B,E}, {A,D})
({C,F}, {E,D}, {A,B})
({E,D}, {A,B}, {C,F})
({E,D}, {A,C}, {B,F})
({E,D}, {A,F}, {C,B})
({E,D}, {C,B}, {A,F})
({E,D}, {B,F}, {A,C})
({E,D}, {C,F}, {A,B})
({D,F}, {A,B}, {C,E})
({D,F}, {A,C}, {B,E})
({D,F}, {A,E}, {C,B})
({D,F}, {C,B}, {A,E})
({D,F}, {B,E}, {A,C})
({D,F}, {C,E}, {A,B})
({E,F}, {A,B}, {C,D})
({E,F}, {A,C}, {B,D})
({E,F}, {A,D}, {C,B})
({E,F}, {C,B}, {A,D})
({E,F}, {B,D}, {A,C})
({E,F}, {C,D}, {A,B})

The $15$ unordered partitions of $S$ into $3$ pairs are

{{A,B}, {C,D}, {E,F}}
{{A,B}, {C,E}, {D,F}}
{{A,B}, {C,F}, {E,D}}
{{A,C}, {B,D}, {E,F}}
{{A,C}, {B,E}, {D,F}}
{{A,C}, {B,F}, {E,D}}
{{A,D}, {C,B}, {E,F}}
{{A,D}, {B,E}, {C,F}}
{{A,D}, {B,F}, {C,E}}
{{A,E}, {C,B}, {D,F}}
{{A,E}, {B,D}, {C,F}}
{{A,E}, {B,F}, {C,D}}
{{A,F}, {C,B}, {E,D}}
{{A,F}, {B,D}, {C,E}}
{{A,F}, {B,E}, {C,D}}

A: $n = 3$
We have $6$ people.
There are ${6\choose 2}$ ways to make the first paring.  That leave 4 left to pair, we pair them up.  And the last pair fall into place.
${6\choose 2}{4\choose 2}{2\choose 2} = \frac {6!}{(2!)(2!)(2!)} = \frac {6!}{2^3}$
But this suggests that the first couple paired is distinct from the second couple paired, and we don't care about these permutations.
$\frac {6!}{2^3(3!)}$
Can you abstract from here to larger groups?
Multiply that out and you get:
