Let $S = \{1, 2,..., 8\}$. How many ways are there to select two disjoint subsets of $S$?
1. Consider the set \( S = \{1, 2, \ldots, 8\} \). We need to determine the number of ways to select two disjoint subsets from \( S \).
2. Each element in \( S \) has three choices:
   - It can be in the first subset.
   - It can be in the second subset.
   - It can be in neither subset.
3. Since there are 8 elements in \( S \) and each element has 3 choices, the total number of ways to assign each element to one of the three categories is given by:
   \[
   3^8
   \]
4. Calculating \( 3^8 \):
   \[
   3^8 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 6561
   \]

The final answer is \( \boxed{6561} \).