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Girls and Boys in the same school, each girl know 10 boys and each boy knows 10 girls, prove that the number of boys and the number of girls in the school is the same (know is symmetric = if girl G is know the boy B, then B is know G)

I think this question is related with Pigeonhole problem, so I was trying to use Math Induction to prove it. But it is more likely about graph theory, can anyone help me about this?

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Draw a bipartite graph, with boys on one side and girls on the other. Draw edge between a girl and a boy if and only if they know eachother. Then each vertex has degree 10.

Question: How many edges are there? Hint Count them from the "boy" vertices, then again from the "girls" vertices...

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  • $\begingroup$ so it should be a direct graph? $\endgroup$ – JavaLeave Jul 8 '13 at 3:34
  • $\begingroup$ @leaveJava Not necessarily, bi-partite is enough. But making it a di-graph might work better, then you can easely count the total in-degree and the total out-degree. $\endgroup$ – N. S. Jul 8 '13 at 3:38

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