Counting Binary Strings

How many binary strings of length $n$ exist? Each position has 2 choices (0 or 1), so by the product rule: $2^n$ strings.

For $n = 3$: 000, 001, 010, 011, 100, 101, 110, 111. That is $2^3 = 8$ strings. Notice this matches the number of subsets of a 3-element set. Each binary string encodes which elements to include.