We should hope someone does come up with such a cure! It would save a lot of sniffling.
First, there is one big problem with this question that makes it essentially unanswerable. There really is no such thing as the "common cold." What we refer to as "a cold" is really a virus that comes in about 200 different strains. That's why you can keep on getting it year after year, despite that your body's immune system has battled it before. It's because your body may not have battled that exact virus. Just because the symptoms are the same doesn't mean the same virus is responsible.
So, if someone claimed a cure for the "common cold" it would be quite incredible. They'd be essentially saying that they could defeat these 200 different viruses and also defeat whatever they'll mutate to in years to come. That's a bold claim.
Were it possible, the second part of the question has to do with the term "random group." There are different types of "random." A random group of people drawn from one part of town isn't going to be as random as a group drawn from two or three different towns. The random 100 might be of a similar age (maybe because of an outbreak at an old-folks home,) race, or sex. You'd have to be certain that the individuals were not just random but also diverse.
Secondly, just because someone has a cold doesn't mean that they are in the same "stage." Unless there are other factors, a person's immune system will defeat the cold on its own. Are the 100 people in the same stage of their cold? That would be factor.
100 people, incidentally, is not enough for a meaningful study. The problem is in the need for a control group. You'd need to have half your group as a control that would get a placebo instead of a real drug. Ideally, you'd have a third group you could monitor that didn't get anything at all, thus defeating the "placebo effect" in which people might think themselves healthy.
You'd have to give the drug to one group, give a fake drug to another, and nothing to a third. Then, you'd have to monitor them. In its simplest form, you'd have those in the study "self report" on the severity of their symptoms, perhaps several times a day. I suppose you could do some blood test for the presence of the virus and work it that way, but the cost would be driven way up.
After a preset amount of time you'd have to compare the three groups and their reporting. Most interesting would be the two groups against the third that got no treatment. Did those that took the treatment report feeling better faster? More interestingly, did those that took the placebo claim they were getting better just as fast?
There's more to it than this, but hopefully I've given you something to work with. In short, there's a reason why nobody has figured out how to kill off colds, and why most people interested in the subject are just trying to manage the symptoms effectively.