For each of n elements, you make a binary choice: include it or exclude it from the subset.
Element 1: 2 choices. Element 2: 2 choices. Continue for all n elements. Total subsets: 2 × 2 × ... × 2 (n times) = 2^n.
This matches the sum of combinations: you can choose 0 items in C(n,0) ways, 1 item in C(n,1) ways, etc. The sum equals 2^n.