Serious applications of a party trick

In a group of 30 people, it's likely that two people have the same birthday. For a group of 23 the probability is about 1/2, and it goes up as the group gets larger.

In a group of a few dozen people, it's unlikely that anyone will have a particular birthday, but it's likely that two people will have the same birthday. This makes a good party trick, but it's much more than a party trick. It illustrates a general principle that comes up frequently in applications.

For example, it explains a common mistake in seeding random number generators and it lets you predict the chances of hash function collisions. Also, simulation of the birthday problem is used as a test for random number generators because the simulation is sensitive to flaws such as correlated output.

What are other examples of mathematical party tricks that illustrate an important principle in applications? I'm sure there are others, but the birthday problem is the first one that comes to mind.