How many distinct ways can the set {1,2,3,4} be partitioned into exactly 2 non-empty subsets?
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The partitions are determined by which subset contains 1. The other subset can be any non-empty proper subset of {2,3,4}: namely {2,3,4}, {2,3}, {2,4}, {3,4}, {2}, {3}, or {4} — giving 7 partitions. (Equivalently, 23−1=7, since we exclude the case where the other subset is empty.)