The question is how to calculate for the general problem.
The specific problem is this:
Given a door code of $n$ numbers on a 10 digit panel, how many combinations do you have to try?
The catch is that the mechanism that senses whether the correct code has been entered has no concept of the start and end of the code. So the sequence 123456 is actually testing 3 unique door codes - my particular case uses a 4 digit code.
The question is, given this modification how many combinations are there given a code length of $n$?
In answer to the comment:
Yes, I am looking for the shortest sequence of digits that tests all combinations.