To find the total number of unordered pairs of disjoint subsets of S={1, 2, 3, 4}, we need to consider all possible combinations of subsets that are disjoint.

Let's break down the problem step by step:

Step 1: Finding the number of subsets of S
The set S has 4 elements, so the total number of subsets can be found by taking the power set of S. The power set of a set with n elements has 2^n subsets. Therefore, the power set of S will have 2^4 = 16 subsets.

Step 2: Identifying the disjoint subsets
For two subsets to be disjoint, they should not share any common elements. In other words, the intersection of the two subsets should be empty.

To form an unordered pair of disjoint subsets, we need to select two subsets from the power set of S such that their intersection is empty.

Step 3: Counting the unordered pairs
To count the unordered pairs of disjoint subsets, we need to find all possible pairs of subsets that are disjoint.

Let's consider the subsets of S:
Subset 1: {}
Subset 2: {1}
Subset 3: {2}
Subset 4: {3}
Subset 5: {4}
Subset 6: {1, 2}
Subset 7: {1, 3}
Subset 8: {1, 4}
Subset 9: {2, 3}
Subset 10: {2, 4}
Subset 11: {3, 4}
Subset 12: {1, 2, 3}
Subset 13: {1, 2, 4}
Subset 14: {1, 3, 4}
Subset 15: {2, 3, 4}
Subset 16: {1, 2, 3, 4}

We can see that the empty set {} cannot form a disjoint pair with any other subset.

Now, let's count the pairs:
- Subset 1 {} can be paired with subsets 2 to 16 (15 pairs)
- Subset 2 can be paired with subsets 3 to 16 (14 pairs)
- Subset 3 can be paired with subsets 4 to 16 (13 pairs)
- Subset 4 can be paired with subsets 5 to 16 (12 pairs)
- Subset 5 can be paired with subsets 6 to 16 (11 pairs)
- Subset 6 can be paired with subsets 7 to 16 (10 pairs)
- Subset 7 can be paired with subsets 8 to 16 (9 pairs)
- Subset 8 can be paired with subsets 9 to 16 (8 pairs)
- Subset 9 can be paired with subsets 10 to 16 (7 pairs)
- Subset 10 can be paired with subsets 11 to 16 (6 pairs)
- Subset 11 can be paired with subsets 12 to 16 (5 pairs)
- Subset 12 can be paired with subsets 13 to 16 (4 pairs)
- Subset 13 can be paired with subsets 14 to 16 (3 pairs)
- Subset 14 can be paired with subsets 15 to 16 (2 pairs)
- Subset 15 can be paired with subset 16 (1 pair)

Therefore, the total number of unordered pairs of disjoint subsets is 15 + 14 +