Animated Solution for Mathematics - Permutations and Combinations: Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to

We are given the set $S=\{1,2,3,4\}$. Our goal is to determine the number of unordered pairs of disjoint subsets $\{P,Q\}$ such that $P\cap Q=\varnothing$.

To solve this, we consider the fate of each individual element $x\in S$. Because the sets $P$ and $Q$ must be disjoint, each element $x$ has exactly three mutually exclusive options:

1. $x\in P$ 2. $x\in Q$ 3. $x otin P$ and $x otin Q$

Since there are 4 elements in $S$ and each element makes its choice independently, we apply the fundamental principle of counting. For each of the 4 elements, there are 3 possible choices.

The total number of ordered pairs $(P,Q)$ is given by:

The problem asks for unordered pairs $\{P,Q\}$, where the pair $\{P,Q\}$ is considered identical to $\{Q,P\}$. In our count of 81, we have included both $(P,Q)$ and $(Q,P)$ as distinct entities whenever $P eq Q$.

However, we must identify the case where $P=Q$. Given the condition $P\cap Q=\varnothing$, the only way $P=Q$ is if both sets are empty:

This specific case, $(\varnothing ,\varnothing )$, is the only instance where the ordered pair does not have a distinct "partner" to form an unordered pair.

We subtract this unique case from the total and divide by 2 to account for the symmetry of the remaining ordered pairs:

Calculating this, we find:

The total number of unordered pairs of disjoint subsets is 41.