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I'm hoping I can find some help to solve a puzzle that came up last night with friends. I thought I could find a solution but have been out of college too long. We had 3 couples over (8 adults), and wanted to play a single round for each possible pairing of the 8 people. After playing with each person as a partner, we' have an individual emerge with the highest overall score.

I thought about 10 rounds would cover each combination, and we simply played until we knew we'd each played with every other person. This got me thinking of my old linear algebra class in college and the 'n choose y' formulas involving factorials.

I've googled and found the combinatorial matrix formula, but can't use it to properly get the number I'm looking for.

Goal: How to apply the C(n,r) formula to properly seat 8 adults into a single pairing with every other member.

My (wrong?) strategy is a 2-pass process:

pass 1: find the number of combinations of all 8 C(8,2) = 28. I know this is correct.

pass 2 (my fail): We must know how many ways to combine each of those 28 pairs into groupings of 4. Isn't this C(28,4)? Obviously not, because that's 20,475. We'd played every combination in about 9 rounds.

Many thanks for anyone willing to help me clarify this!

For reference, the combinational matrix formula:

C(8,2) = 8! / 2! (8 - 2)! = 28

C(28,4) = 28! / 4! (28 - 4)! = 20475

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# Euchre Tournament random seating algorithm

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