24.1. (Czechoslovakia, 73). How many different pairs of non-intersecting subsets does a set consisting of $n$ elements have?
24.1. Let's find the number of different pairs of non-intersecting subsets under the condition that the first and second subsets are distinguished in the pair. For each of the $n$ elements, there are 3 possibilities: it can either be included in the first subset, or included in the second subset, or not included in either of them. Therefore, the number of such pairs is $3^{n}$. Among them, there is one pair where both subsets are empty. The remaining ( $3^{n}-1$ ) pairs can be divided into pairs of identical pairs if we allow swapping the first and second subsets in the pairs. Thus, there are ( $\left.3^{n}-1\right) / 2$ (unordered) pairs of subsets, where at least one is not empty. In total, there are $\left(3^{n}-1\right) / 2+1=\left(3^{n}+1\right) / 2$ different pairs of subsets that satisfy the condition of the problem.