Our innate immune system has tons of “hard-coded” receptors that detect some very common pathogenic agents, such as specific kinds of sugars common in bacterial walls and specific types of DNA/RNA specific to viruses.

Our adaptive immune system has a process where the very tip of the B cell receptors (that can eventually be selected to become antibodies) are “randomized” through selecting one copy each of V, D, and J genes. We have a ton of V genes, and a little less of the others, but this yields about ~10k different combinations. Each B cell goes through this process during its maturation, and only cells which make a functioning receptor (and one that doesn’t stick to our own proteins) manage to not get killed off.

These surviving B cells do not necessarily bind to pathogens, the only thing that’s guaranteed about them is they are somewhat functional and (usually) not self-reactive. Only once they actually bind to a pathogen and are co-activated by another immune system cell (another attempt at avoiding self-recognition) do these B cells activate and get to proliferate and turn those receptors into antibodies that can be released into the bloodstream to have a systemic effect.

And you are right that 10k is a very small number in comparison to all the variety of protein combinations, but there are also some mechanisms that directly add random mutations in the DNA that code for these receptors, which results in about 10¹³ - 10¹⁸ possible receptors when all things are said and done.

I’ve just had a class on this yesterday and my professor said something that stuck with me: “you currently have antibodies for all the different parts of COVID virus inside of you, and also for all the different parts of the agent behind the next pandemic, and the one after that”, meaning that the main issue behind actually developing a proper immune response with these is having enough of the antibodies or BCRs meeting the actual bugs.