Why are there only 2^i configurations?
I don't know which proof you're referring to, but i'd expect it's because the input is assumed to be binary. I.E. a binary string of length 1 can have two values, 0 or 1. A binary string of length 2 can have 4 values, 00,01,10,11, and a binary string of length 3 can have 8 values, and so on.