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 is a subset of T?

Correct Answer: The correct option is 729.

To find the number of pairs of subsets (S,T) of the set {1,2,3,4,5,6} such that S⊆T, we can analyze the choices available for each element in the set.

Method 1: Element-wise Analysis
Let U={1,2,3,4,5,6} be the universal set, which contains n=6 elements.
For any element x∈U and any pair of subsets (S,T) satisfying S⊆T, there are exactly three mutually exclusive possibilities:
1. The element x belongs to neither S nor T (i.e., x∉T and x∉S).
2. The element x belongs to T but does not belong to S (i.e., x∈T and x∉S).
3. The element x belongs to both S and T (i.e., x∈T and x∈S).
Note that the case where x∈S but x∉T is impossible because S must be a subset of T.

Since each of the 6 elements has exactly 3 possible placement choices independent of the other elements, the total number of valid pairs (S,T) is:
36=3×3×3×3×3×3=729

Method 2: Inductive Analysis (as shown in the provided image)
Let us verify the pattern by starting with simpler sets of fewer elements:
• For 1 element (e.g., U = {1}):
- If T=∅, then S=∅ (gives 1 pair: (∅,∅))
- If T={1}, then S=∅ or S={1} (gives 2 pairs: (∅,{1}) and ({1},{1}))
Total pairs for 1 element = 1+2=3=31

Following this induction pattern:
• For 2 elements: total pairs = 32=9
• For 3 elements: total pairs = 33=27
• For 4 elements: total pairs = 34=81
• For 5 elements: total pairs = 35=243
• For 6 elements: total pairs = 36=729