Problem:
How many subsets of contain at least one prime number?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The number of qualifying subsets equals the difference between the total number of subsets of and the number of such subsets containing no prime numbers, which is the number of subsets of . A set with elements has subsets, so the requested number is .
A subset meeting the condition must be the union of a nonempty subset of and a subset of . There are of the former and of the latter, which gives choices in all.
The problems on this page are the property of the MAA's American Mathematics Competitions