MathNet 2002 — Part 01 Question 33

For any non-empty subset $A$ of $\{1, 2, \dots, n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$. Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \dots, n\}$.