How many subsets of $\{0,1,\ldots,9\}$ have the property that there are at least two elements and the sum of the two largest elements is $13$? (For example, $\{1,4,9\}$ is such a subset.)
I'm having a hard time to figure out the ways to attack this problem ... casework would take too long, and complementary counting seems just as much.
Let $a<b =13-a$ be largest elements in the set.
If $b=9$ then $a=4$ so we can choose any subset of $\{0,1,2,3\}$ so we have $2^4=16$ subsets.
If $b=8$ then $a=5$ so we can choose any subset of $\{0,1,2,3,4\}$ so we have $2^5=32$ subsets.
If $b=7$ then $a=6$ so we can choose any subset of $\{0,1,2,3,4,5\}$ so we have $2^6=64$ subsets.
So we have $16+32+64 =112$ sets.