Think about this problem: Given $S=(a, b, c, d)$, in how many ways can we choose two subsets $A, B$, such that $S=A \cup$
$$
B \text { ? }
$$
Analysis: Generally, the methods we commonly use are case analysis, and we can also use knowledge related to permutations and combinations.
Solution: Let subsets $A$ and $B$ satisfy the condition, then each element in set $S$ either belongs only to $A$, only to $B$, or to the intersection of $A$ and $B$, and except for $A=B=S$, each set is counted twice, so there are $\frac{3^{4}+1}{2}=41$ ways.
Comment: We analyze the problem from a different perspective, avoiding the case-by-case discussion based on the number of elements in the subsets.