Let's take human DNA as an example. It contains 3.2B GTCA base pairs. This gives rise to 4^3.2B possible combos. It's just not possible to navigate this space blindly. There is not enough atoms in the universe to do that. It is known that there is bias in what mutations are favoured.

Only a tiny percentage (around 1%) of the DNA in chromosomes codes for proteins.

And yes, certain mutations are favored precisely because of the chemistry constraints (an extremely basic one is which base pair changes actually alter the resulting protein; a more sophisticated one is which amino acid changes alter the physical functionality of the protein).