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In a hypothetical game, 2 numbers of nine wins. If I want to increase my chances, I would use combinations of 3. There is 84 combinations of 9 numbers (1 to 9) 3 to 3, but only 12 combinations of 3 groups all combinations of 9 2 to 2: 1 2 3, 1 4 7, 1 5 9, 1 6 8, 2 4 9, 2 5 8, 2 6 7, 3 4 8, 3 5 7, 3 6 9, 4 5 6, 7 8 9,

There is some method, algorithm, technique, whatever, that I can use to list all 8407 combinations of 50 numbers (1 to 50) taken 10 to 10, where 5 numbers wins? There is 2118760 combinations of 50 5 to 5, but only 8407 combinations of 50 10 to 10, to hit 5. Can anyone please help me to LIST the combinations?

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Could you please explain more clearly how this hypothetical game works? From what I understand you want to construct a set of triples from $\{1, \ldots, 9\}$ such that every pair $\{x, y\}$ occurs together in one of these triples. –  TMM Sep 7 '11 at 1:45
    
That´s right! In my question, I list all 12 combinations, but how to do if I have much more numbers, as a lottery with 50 numbers where 5 wins? I want to try some combination of 50 10 to 10 to win 5, to increase chances... –  GuyBR Sep 7 '11 at 1:54
    
This is a lottery wheel or cover design. You can find some rather old links here –  Henry Sep 7 '11 at 7:17

1 Answer 1

This type of problem is a specific instance of a covering design. A covering design is a set of three numbers usually expressed in the form $(v,k,t)$. This describes a collection of "blocks" of $k$-member elements chosen from a set of cardinality $v$ such that each $t$-member subset of $v$ appears at least once.

This website is an online collection of such designs where you can input your own $(v,k,t)$ and retrieve previously created covers that meet those parameters.

The only covering design in their archive with $v=50$, $k=10$, and $t=5$ can be found here. Unfortunately, this particular design has more than twice as many blocks as the OP wanted.

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That final link actually has 19831 lines, and its title suggests at least 8820 are needed. –  Henry Sep 7 '11 at 6:51
    
@Henry It was the only one the had for those parameters, but I didn't think to actually look at the number of blocks. Edited answer. –  user12998 Sep 7 '11 at 14:03

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