I was wondering why the flu vaccination doesn't protect us from all different types of flu. I know there are 3 major groups A, B and C and they mutate really fast. For example Influenza A virus has 2 different proteins on the surface hemagglutinin (H) and the neuraminidase (N). There are 18 different hemagglutinin subtypes and 11 different neuraminidase subtypes. So if they produce a vaccination which covers 187 of the virus (from H1N1 to H18N11) wouldn't it mean that the Influenza Type A could be eradicated from the world ? And wouldn't it work the same way for Type B and C?
I'm going to try and focus on the latter part of your question; if you want to know why the tri- or quadrivalent flu vaccine doesn't provide protection like many other vaccines, that's already been answered.
Anyway, the main reason we can't is because there are simply too many. You say 187 (not sure why,
18*11=198) but the truth is that not all H1s are the same. The different subtypes constantly mutate, so one H1N1 may not be the same as another H1N1. There are hundreds upon hundreds of flu types, constantly changing. And that's not even accounting for type B viruses.
When you make a vaccine, you need a certain amount of protein. The major quadrivalent vaccine study used 15µg of protein of each strain. The more protein you put in, the bigger the vaccine, the more it hurts, and the more likely the body is to have a significant negative reaction. You'd have to test all of them as well - the FDA doesn't usually work on a "three is safe, four is safe, so hundreds is safe" principle.
Plus, you have to MAKE all that protein. We're not very good at making enough for the tri- or quadrivalent vaccines now, much less hundreds or thousands of times more. We simply don't have the capacity for that level of production. Even if we could, it's not likely that immunity to all strains would take. Besides, erradication is unlikely because there are large animal flu reservoirs (avian and swine, most famously) that can and will jump over to humans. That's the piece everyone forgets about smallpox - with no animal reservoir, it was easy to eradicate.
There are 18 different hemagglutinin subtypes and 11 different neuraminidase subtypes that have been reported! The point is that many more are present in the overall viral population and some variant will escape the effect of the vaccination. In general however, to cover multiple variants with the same vaccination sounds a good idea.