A certain company gives each of its clients a 10 digit number as a sort of identification code. As a precaution, any pair of used codes should differ by at least two digits so no one accidentally gives someone else's code.
How many clients can they have before adding digits? Give an example of a set of codes they might use. What if each pair of codes must differ by at least 3 digits? 4? More?
They can have 10^9 clients. This works if, for example, the sum of all the digits mod 10 is zero for each code.
For 10 digit identification code, with any pair differing by at least two digits, the number of clients is 1000,000,000.