Let's move from the bathroom into the kitchen, since that's a socially acceptable location to have more than one person at the same time. In the kitchen, you may not want to have everyone in there at once. In fact, you probably want to limit the number of people you can have in the kitchen (too many cooks, and all that).
Let's say you don't ever want to have more than two people in there simultaneously. Could you do it with a mutex? Not as we've defined it. Why not? This is actually a very interesting problem for our analogy. Let's break it down into a few steps.
The bathroom can have one of two situations, with two states that go hand-in-hand with each other:
No other combination is possible—the door can't be locked with nobody in the room (how would we unlock it?), and the door can't be unlocked with someone in the room (how would they ensure their privacy?). This is an example of a semaphore with a count of one—there can be at most only one person in that room, or one thread using the semaphore.
The key here (pardon the pun) is the way we characterize the lock. In your typical bathroom lock, you can lock and unlock it only from the inside—there's no outside-accessible key. Effectively, this means that ownership of the mutex is an atomic operation—there's no chance that while you're in the process of getting the mutex some other thread will get it, with the result that you both own the mutex. In our house analogy this is less apparent, because humans are just so much smarter than ones and zeros.
What we need for the kitchen is a different type of lock.
Suppose we installed the traditional key-based lock in the kitchen. The way this lock works is that if you have a key, you can unlock the door and go in. Anyone who uses this lock agrees that when they get inside, they will immediately lock the door from the inside so that anyone on the outside will always require a key.
Well, now it becomes a simple matter to control how many people we want in the kitchen—hang two keys outside the door! The kitchen is always locked. When someone wants to go into the kitchen, they see if there's a key hanging outside the door. If so, they take it with them, unlock the kitchen door, go inside, and use the key to lock the door.
Since the person going into the kitchen must have the key with them when they're in the kitchen, we're directly controlling the number of people allowed into the kitchen at any given point by limiting the number of keys available on the hook outside the door.
With threads, this is accomplished via a semaphore. A “plain” semaphore works just like a mutex—you either own the mutex, in which case you have access to the resource, or you don't, in which case you don't have access. The semaphore we just described with the kitchen is a counting semaphore—it keeps track of the count (by the number of keys available to the threads).