2017 AIME I Problems/Problem 12
Call a set product-free if there do not exist (not necessarily distinct) such that . For example, the empty set and the set are product-free, whereas the sets and are not product-free. Find the number of product-free subsets of the set .
We shall solve this problem by doing casework on the lowest element of the subset. Note that the number cannot be in the subset because . Let be a product-free set. If the lowest element of is , we consider the set . We see that 5 of these subsets can be a subset of (, , , , and the empty set). Now consider the set . We see that 3 of these subsets can be a subset of (, , and the empty set). Note that cannot be an element of , because is. Now consider the set . All four of these subsets can be a subset of . So if the smallest element of is , there are possible such sets.
If the smallest element of is , the only restriction we have is that is not in . This leaves us such sets.
If the smallest element of is not or , then can be any subset of , including the empty set. This gives us such subsets.
So our answer is .
Solution 2(PIE) (Should be explained in more detail)
We cannot have the following pairs or triplets: . Since there are = subsets( isn't needed) we have the following: The total number of sets with at least one of the groups (with repeats) = = The total number of sets with at least two of the groups (with repeats) = 1st & 2nd pair 1st & 3rd pair 1st & 4th pair 2nd & 3rd pair 2nd & 4th pair 3rd & 4th pair = = The total number of sets with at least three of the groups (with repeats) = 1st, 2nd, & 3rd groups 1st, 2nd & 4th groups 1st, 3rd, & 4th groups 2nd, 3rd, & 4th groups = = The total number of sets with all of the groups. = = . (Note: If someone can add breaks and maybe help explain why there are powers of 2 that would be good)
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