Counting techniques to improve efficiency I am having trouble with the following question:

There are four people named W,X,Y,Z. Each person is currently in a room and each room can only hold one person. If we randomly shuffle these four people between different rooms, what is the probability that exactly two of these people will end up in the same room where they started in the beginning. 

I let the event $A$ represent two people returning to the same rooms. I let $n(A)$ be the number of ways this event can happen. I am able to calculate $n(A)$ but I am looking for a more efficient way to do so.
I let $n(A) = (1\times3\times1\times1) + (3\times1\times1\times1)$
since I realized that the outcomes would depend on whether the first person entered the correct room. If the number of people and rooms increased this method would take too long.
 A: In order for exactly two of the people to end up in their original rooms, the shuffle must simply have switched the other two people. For instance, if $W$ and $X$ end up in their original rooms, $Y$ and $Z$ have simply switched rooms. There’s exactly one shuffle that leaves $W$ and $X$ as the only people in their original rooms. Thus, all you have to do is note that there are $\binom42=6$ pairs of people: for each pair there is one shuffle leaving that pair fixed and interchanging the other pair. 
This is the most efficient way when there are $4$ people, but it won’t generalize. The general problem is a bit nasty. Suppose that you have $n$ people. There are $\binom{n}2$ ways to choose the $2$ people who get their original rooms. The shuffle must then move each of the other $n-2$ people to a different room. In other words, you need a derangement of the other $n-2$ people. The link gives formulas for the number of derangements; perhaps the nicest from a purely computational point of view is that the number of derangements of a set of $n$ things is
$$\left[\frac{n!}e\right]=\left\lfloor\frac{n!}e+\frac12\right\rfloor\;,$$
where $[x]$ is the integer nearest $x$. The article also gives a couple of useful recurrences, and there is more information, with references, in the OEIS entry for the sequence of derangement numbers.
