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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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    This is a *[lottery wheel](http://en.wikipedia.org/wiki/Lottery_Wheeling)* or *cover design*. You can find some rather old links [here](http://thelotteryinstitute.tripod.com/wheels.htm)2011-09-07

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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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    @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.2011-09-07