Count number of subset

Question

For this problem, I find it time-consuming to methodically go through writing down elements of each $S_i$ and check if there are any identical sets of apparently different expression. Are there any way to see this fast without doing much enumeration? Answer is (D) by the way.

Answer 1

Note that $S$ is contained in $\sigma(A, B)$ (the $\sigma-algebra$ generated by {A, B} ⊆ $P(M)$).
As |$\sigma(A_1, ..., A_k)$| $\leq$ |$P(P(1, ..., k))$| $ = 2^{2^k} $ (look here for details), we conclude that
|$S$| $\leq$ |$\sigma(A, B)$| $\leq 16$.

Now take $M = \{1, 2, 3, 4\}$, $A = \{1, 2\}$, and $B = \{2, 3\}$.
Can you see why |$S$|$ = 16$ in this case?