Question

How many pairs of sets \((S, T)\) are possible among the subsets of \(\{1,2,3,4,5,6\}\) that satisfy the condition that \(S \subseteq T\)?

• A. 729
• B. 728
• C. 665
• D. 664

Hint:

When counting subsets with \(S \subseteq T\), think element-wise. Each element has 3 options (in T only, in both S and T, or in neither).

Answer 1

The Correct Option is A

Step 1: Elements and subsets.
The set has 6 elements: \(\{1,2,3,4,5,6\}\). Number of subsets = \(2^6 = 64\).

Step 2: Condition \(S \subseteq T\).
For each element, there are 3 possibilities: 1. The element is in \(T\) but not in \(S\).
2. The element is in both \(S\) and \(T\).
3. The element is in neither \(S\) nor \(T\). Thus, each element has 3 valid choices.

Step 3: Total number of pairs.
\[ \text{Total pairs} = 3^6 = 729 \] \[ \boxed{729} \]