Permutations and combinations - distributing objects into groups Question:

In how many ways can $2n$ people be divided into $n$ pairs?

Approach:
Well as there are $2n$ people, it is obvious that we need to chose each and everyone of them. Using simple counting, the way to divided $2n$ people into $n$ groups is $(2n)!$ However, there is the chance that some of the pairs are repeated. How exactly would I proceed to answer this question?

Here is another approach you could use, which gives the same answer:
1) Take an arbitrary person in the group;  there are $2n-1$ choices for this person's partner.
2) Next take anyone in the remaining $2n-2$ people; there are $2n-3$ choices for this person's partner.
3) Now take anyone in the remaining $2n-4$ people; there are $2n-5$ choices for the partner of this person.
Continuing in this manner, 
there are $(2n-1)(2n-3)(2n-5)\cdots3\cdot1$ ways to divide the group of $2n$ people into pairs.
