MathDB
Minimally Intersecting

Source: AIME 2010I Problem 7

March 17, 2010
AMCAIME IAIME

Problem Statement

Define an ordered triple (A,B,C) (A, B, C) of sets to be minimally intersecting if |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1 and A \cap B \cap C \equal{} \emptyset. For example, ({1,2},{2,3},{1,3,4}) (\{1,2\},\{2,3\},\{1,3,4\}) is a minimally intersecting triple. Let N N be the number of minimally intersecting ordered triples of sets for which each set is a subset of {1,2,3,4,5,6,7} \{1,2,3,4,5,6,7\}. Find the remainder when N N is divided by 1000 1000. Note: ∣S∣ |S| represents the number of elements in the set S S.
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