Optimizing number of 6-digit strings differing in at least two places 
A certain province issues license plates consisting of six digits (from 0 to 9). The province
  requires that any two license plates differ in at least two places. (For instance, the numbers
  $027592$ and $020592$ cannot both be used.) Determine, with proof, the maximum number of
  license plates that the province can use.

I have seen a few similar questions in the past, but I have never been able to solve these optimization problems.
My idea is to divide all the permutations into sets, and then show that if we pick one permutation from one set, then we can't have more than one permutation from that set.
As with many optimization problems, we also have to show that a construction exists. This is the part I am usually more comfortable with, and I realize that our choice of permutations should be fairly symmetric. I am trying to use basic number theory to create a construction but can't seem to find any.
So, how do I even start this problem?

The province can use up to $10^5$ license plates.
To see that this many license plates can be used, fix some residue $r$ modulo $10$ and consider all license plates whose digit sum has remainder $r$ modulo $10$. Any two of them differ in at least two digits, since changing one digit changes the remainder.
To see that not more license plates than this can be used, note that any two license plates must differ in the last $5$ digits.
