Problem:
Find the number of subsets of {1,2,3,4,5,6,7,8} that are subsets of neither {1,2,3,4,5} nor {4,5,6,7,8}.
Solution:
There are 28=256 subsets of {1,2,3,4,5,6,7,8}. Each of the sets {1,2,3,4,5} and {4,5,6,7,8} has 25=32 subsets, and their intersection, {4,5}, has 22=4 subsets. Thus the number of subsets of either {1,2,3,4,5} or {4,5,6,7,8} is 32+32−4=60, and the number of subsets of {1,2,3,4,5,6,7,8} with the required property is 256−60=196​.
The problems on this page are the property of the MAA's American Mathematics Competitions