# Does a 10-element set have 30 3-element subsets such that each pair is in two of these 30 subsets?

Does a 10-element set have 30 3-element subsets such that each pair is in two of these 30 subsets?

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Dear David. I fear that some will tend to think that your question is a homework, and will therefore vote to close it. To avoid that, you should indicate a little bit the context in which your questions arose. In particular, where to the numbers 10 and 30 come from? – André Henriques Jun 18 '12 at 20:28
I light of Douglas Zare's answer, this question did not deserve to be closed. I vote to reopen. – André Henriques Jun 18 '12 at 21:51
I still think the question would benefit from some explanation as to the OP's motivation. It isn't the kind of question that pops into my head at random, although I don't claim to be representative. – Yemon Choi Jun 19 '12 at 0:17
This question was motivated by communications scheduling. It concerns a n-node network in which up to m nodes can broadcast at a time; nodes can not receive while sending, but when not sending they can simultaneously receive from all m senders. The goal is to serve all source-destination pairs equally. n=10 and m=3 are arbitrary choices. 30 is the minimum possible because there are n(n - 1) = 90 pairs, n(n - m) = 21 can be served at a time, and lcm(90, 21)/21 = 30. I was assuming the 30 subsets are distinct, but now I see that my application doesn't need that. – David Wasserman Jun 19 '12 at 16:48

This asks for a $(10,3,2)$ balanced incomplete block design. These are known. There are $960$ different designs with those parameters up to isomorphism according to the CRC Handbook of Combinatorial Designs.