24. [12] Find the number of subsets $S$ of $\{1,2, \ldots 6\}$ satisfying the following conditions:
- $S$ is non-empty.
- No subset of $S$ has the property that the sum of its elements is 10 .
Answer: $\quad 34$ We do casework based on the largest element of $S$. Call a set $n$-free if none of its subsets have elements summing to $n$.
Case 1: The largest element of $S$ is 6 . Then $4 \notin S$. If $5 \notin S$, then we wish to find all 4 -free subsets of $\{1,2,3\}$ (note that $1+2+3=6<10$ ). We just cannot include both 1,3 , so we have $2\left(2^{2}-1\right)=6$ choices here.
If $5 \in S$, then we want 4,5 -free subsets of $\{1,2,3\}$. The only 4 -but-not- 5 -free subset is $\{2,3\}$, so we have $6-1$ choices here, for a case total of $6+5=11$.

Case 2: The largest element of $S$ is 5 . We seek 5,10 -free subsets of $\{1,2,3,4\}$. We just cannot have both 1,4 or both 2,3 (note that getting 10 requires the whole set), so we have $\left(2^{2}-1\right)^{2}=9$ subsets in this case.
Case 3: The largest element of $S$ is at most 4 . (So we want a 4 -free subset of $\{1,2,3,4\}$.) The only way to sum to 10 with $1,2,3,4$ is by using all the terms, so we simply discount the empty set and $\{1,2,3,4\}$, for a total of $2^{4}-2=14$ subsets.
In conclusion, the total number of subsets is $11+9+14=34$.