Discussion
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?
Answer: A
There are 729 such pairs.
Decide, element by element, where each of the six numbers goes when S is a subset of T. Each element has exactly three possibilities: it is in neither set, it is in T but not in S, or it is in both S and T. (It cannot be in S without being in T.) The choices are independent, so
\[ 3^{6} = 729 \]
728 leaves out one pair, such as the pair of two empty sets, which is a valid pair since the empty set is a subset of itself. 665 is \(729 - 64\), which drops the 64 pairs where S equals T; but 'S is a subset of T' includes equality. 664 makes both of those deductions.
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