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I have a (probably) simple problem, but due to my dusty knowledge, I don't even know where to start reading or how to name my problem.

Our team has grown a lot in the last few weeks and it is very difficult to get to know each other. Therefore I want to make a list with meetings of 3 people each to establish contact. Now I would like to depict all possible encounters under the following conditions:

  • Number of persons/meeting = 3
  • Only people from different sub-teams should meet in each meeting (i.e. max. 1 member from three different sub-teams each).
  • Team size = 21 people in 5 sub-teams
  • Size of the sub-teams varies (4,7,5,3,2 members)

How can I solve the problem? What are such problems called? (Bonus: Where can I read in on this?)

Posted also on Cross Validated

dwightkschruteIII
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  • You ask for "As few as possible encounters of people from the same sub-team". That seems to imply that you want at most one person from each sub-team. Is that a correct interpretation? – awkward Nov 06 '20 at 13:28
  • Thank you very much for asking. This interpretation makes sense. It was important to me that members from the same sub team do not meet each other (they have enough contact with each other). A meeting with two members from one sub-team and one person from another sub-team would be fine, but it doesn't really make much sense. I will adapt it in the post. – dwightkschruteIII Nov 06 '20 at 13:52
  • With that interpretation, there are $651$ possible meetings. Are you sure you want a list? Seems a bit long to be useful. – awkward Nov 06 '20 at 15:28
  • You're right. How exactly do you come up with the number? And what would you suggest under the objective? – dwightkschruteIII Nov 06 '20 at 17:20

1 Answers1

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First, let's see if we can calculate how many meetings of three people are possible, with at most one member from each sub-team. Consider the polynomial $$f(x) = (1+4x)(1+7x)(1+5x)(1+3x)(1+2x)$$ and expand, with result $$f(x)=1+21 x+169 x^2+651 x^3+1198 x^4+840 x^5$$ The coefficient of $x^n$ in this result is the number of meetings that can be formed with $n$ people, at most one from each sub-team. In particular, the coefficient of $x^3$ is the number of possible meetings with three people: $651$. This seems too long a list to be useful, but maybe we can say something about ways to pick people for meetings.

One way to proceed is to pick three distinct sub-teams. This can be done in any one of $\binom{5}{3} = 10$ ways. Then pick one person from each of the three sub-teams. If you were to pick the first three sub-teams, for example, then three people, one from each sub-team, can be chosen in $4 \times 7 \times 5 = 140$ ways; so you have plenty of choices. The $10$ ways to pick three sub-teams, if the sub-teams are numbered $1$ through $5$, are $$(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5), \\ (1, 4, 5), (2, 3, 4), (2, 3, 5), (2, 4, 5), \text{ and } (3, 4, 5)$$

I suppose this is obvious, but I don't know what more to say.

($f(x)$, above, is an example of a generating function. Readers interested in learning about generating functions may find many resources in the answers to this question: How can I learn about generating functions?)

awkward
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