Distributing gifts so that everybody gets at least one

As you observed, there are $12^{20}$ ways to distribute the presents without restriction. There are $\binom{12}{k}$ ways to exclude $k$ of the recipients from receiving a present and $(12 - k)^{20}$ ways to distribute the presents to the remaining $12 - k$ people. By the Inclusion-Exclusion Principle, the number of ways to distribute the presents so that each person receives at least one is $$\sum_{k = 0}^{12} (-1)^k\binom{12}{k}(12 - k)^{20}$$