Let S={1,2,3,4}. The total number of unordered pairs of disjoint subsets of S is equal to?

Given, S={1,2,3,4}
Each element can be put in 3 ways: either in the first subset, in the second subset, or in neither subset.
Total number of unordered pairs of disjoint subsets = 3n + 1/2 = 34 + 1/2 = 81 + 1/2
This calculation is incorrect. The correct approach is as follows:
Each element in S can be assigned to one of three possibilities:

1. It belongs to the first subset.
2. It belongs to the second subset.
3. It belongs to neither subset.

Since there are 4 elements in S, there are 34 ways to assign the elements to the subsets. However, this counts ordered pairs. To account for unordered pairs, we must divide by 2 and add back the cases where both subsets are empty (there's only one such case).
However, a simpler approach is to consider the number of ways to partition a set of n elements into three subsets A, B, and C. Each element can be assigned to A, B, or C, so there are 3n ways to do this. In our case, A represents the first subset, B the second subset, and C the remaining elements. Since we are interested in unordered pairs, the total number of such pairs is (3n + 1)/2.
Therefore, the total number of unordered pairs of disjoint subsets is (34 + 1)/2 = (81 + 1)/2 = 82/2 = 41