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15 boys and 15 girls

Source: Russian 2007

September 6, 2007
linear algebramatrixinductiongraph theorycombinatorics proposedcombinatorics

Problem Statement

In the class, there are 15 15 boys and 15 15 girls. On March 8 8, some boys made phone calls to some girls to congratulate them on the holiday ( each boy made no more than one call to each girl). It appears that there is a unique way to split the class in 15 15 pairs (each consisting of a boy and a girl) such that in every pair the boy has phoned the girl. Find the maximal possible number of calls.