Let S={1,2,3,4}. The total number of unordered pairs of disjoint subsets of S is equal to (a) 25

step 1 In this question we are given that S is a set of the four numbers 1, 2, 3 and 4 and we have to c

Let S={1,2,3,4}. The total number of unordered pairs of...

Key Concepts

Ordered versus Unordered Pairs

Distinguishing between ordered and unordered pairs is important in counting problems because it affects the total count. In an ordered pair, the sequence matters, while for an unordered pair the sequence is irrelevant, which often requires adjusting the count by dividing by a factor to correct for overcounting symmetric cases.

Multiplication Principle

The multiplication principle is a basic technique in combinatorial enumeration which involves multiplying the number of independent choices available for each element. For each element in the set, if there are several mutually exclusive categories it can belong to, the total number of assignments is found by taking the product of the number of choices for each element.

Symmetry and Overcounting Corrections

When counting combinatorial objects, it is common to initially count more configurations than necessary due to symmetrical duplicates. Recognizing and correcting for these duplicates is key to arriving at the correct final count, especially when the problem demands an unordered count where different arrangements are considered equivalent.

Power Set

The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself. Understanding power sets is fundamental in combinatorics because it lays the groundwork for counting how many ways elements can be grouped into subsets or pairs of subsets.

Disjoint Subsets

Disjoint subsets are pairs of subsets that do not share any common elements. Recognizing that the subsets must be disjoint is crucial because it imposes a restriction on how elements of the original set are assigned, typically reducing the overall number of combinations available.

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