What makes you think that’s a ridiculously big number? When you’re dealing with numbers outside our usual range of experience, intuition is a very poor guide.
As @Lynn_Munter, there’s no simple answer. But we do have some information to give us clues. Antibodies have come up here before, and they’re a good test case. They’re proteins, and their function is to bind foreign proteins, specifically proteins on invading microbes. Each new microbe encountered has a modest number of potential binding sites, so the task is pretty specific: come up with a protein that will bind one of those sites so you don’t die from an infection. Your immune system routinely comes up with antibodies that do the job, by starting with a stock of ~10^11 random antibodies and varying the ones that have some binding affinity to the target. That must mean that something like 1 in 10^10 random antibodies binds the target well enough to be selected. The variable region of an antibody is about 120 amino acids. I don’t know if they’re all free to vary. If they are, then the number of possible antibodies that will bind to a particular target is ~10^146, i.e. 67 orders of magnitude larger than the ridiculously big number you picked. (If the number of variable sites is smaller, than the absolute number of possible functional proteins is smaller – but the fraction of functional ones, which is the important thing, is unchanged.) Similarly, I’ve seen estimates that 1 in 10^8 random antibodies will have weak catalytic activity for a particular target.