2017 AIME II Problems/Problem 1
Find the number of subsets of that are subsets of neither nor .
The number of subsets of a set with elements is . The total number of subsets of is equal to . The number of sets that are subsets of at least one of or can be found using complementary counting. There are subsets of and subsets of . It is easy to make the mistake of assuming there are sets that are subsets of at least one of or , but the subsets of are overcounted. There are sets that are subsets of at least one of or , so there are subsets of that are subsets of neither nor . .
Upon inspection, a viable set must contain at least one element from both of the sets and . Since 4 and 5 are included in both of these sets, then they basically don't matter, i.e. if set A is a subset of both of those two then adding a 4 or a 5 won't change that fact. Thus, we can count the number of ways to choose at least one number from 1 to 3 and at least one number from 6 to 8, and then multiply that by the number of ways to add in 4 and 5. The number of subsets of a 3 element set is , but we want to exclude the empty set, giving us 7 ways to choose from or . We can take each of these sets and add in a 4 and/or a 5, which can be done in 4 different ways (by adding both, none, one, or the other one). Thus, the answer is .
This solution is very similar to Solution . The set of all subsets of that are disjoint with respect to and are not disjoint with respect to the complements of sets (and therefore not a subset of) and will be named , which has members. The union of each member in and the subsets of will be the members of set , which has members.
Solution by a1b2
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