【Example 10】There are $2 n$ people participating in telegraph training, and every two people form a pair to send and receive messages to each other. How many different pairing methods are there?
Here, we only care about who pairs with whom, similar to the previous example (3), but there is no order between pairs, which differs from the previous example (3). Later, we will introduce that this type of problem belongs to the pairing problem. Now, we will use the perspective of "selecting with order" from permutation problems to provide a solution. Let the $2n$ people line up in a row at random. First, select a partner for the person at the far left, with $2n-1$ ways to choose; then, select a partner for the person at the far left among the remaining people, with $2n-3$ ways to choose; then, select a partner for the person at the far left among the remaining people, with $2n-5$ ways to choose; continue this process until all people are paired. Therefore, there are
$$
(2 n-1)(2 n-3) \cdots 3 \cdot 1=\frac{(2 n)!}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot 2 n}=\frac{(2 n)!}{2^{n} \cdot n!}
$$

different ways to pair them.