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## Main Question or Discussion Point

Hi, sorry if this is in the wrong section. Some of the stuff in this section is way over my head anyway.

I have 10 sets of 3 numbers ranging from 1 to 10. They are interesting in that each number appears three time, no number appears twice in the same set, and no two numbers appear together in different sets more than once.

{1,2,7}

{1,8,0}

{1,5,6}

{3,7,0}

{5,7,9}

{4,5,0}

{2,3,8}

{2,6,9}

{4,6,8}

{3,4,9}

If you took any combination of two of the sets, how many "unique" numbers would you have? In this case a unique number is just one that appears in the sets, so the first two {1,2,7} and {1,8,0} would have unique numbers {1,2,7,8,0}. As it turns out, of the 45 possible combinations of two sets, 30 contain 5 uniques, and 15 contain 6 uniques. I know this because I simulated it.

My question is, is there any way to tell the number times a certain amount of unique numbers will show up in the set without calculating each combination? 30 and 15 are so...symmetric to 45 that I have to imagine there's some easier way to do this. I know that doing this for combinations of 2 sets of three numbers from 1-10 looks easy (and it is), but I'm hoping that by looking at this, there's a more general formula for which you can determine the amount of uniques with combinations of p sets of q parts up to r. For instance, how many different combinations of 50 sets of 5 numbers ranging from 1-1000 will have exactly 200 unique numbers?

I hope someone can either help me out or point me in the right direction. Many thanks.

I have 10 sets of 3 numbers ranging from 1 to 10. They are interesting in that each number appears three time, no number appears twice in the same set, and no two numbers appear together in different sets more than once.

{1,2,7}

{1,8,0}

{1,5,6}

{3,7,0}

{5,7,9}

{4,5,0}

{2,3,8}

{2,6,9}

{4,6,8}

{3,4,9}

If you took any combination of two of the sets, how many "unique" numbers would you have? In this case a unique number is just one that appears in the sets, so the first two {1,2,7} and {1,8,0} would have unique numbers {1,2,7,8,0}. As it turns out, of the 45 possible combinations of two sets, 30 contain 5 uniques, and 15 contain 6 uniques. I know this because I simulated it.

My question is, is there any way to tell the number times a certain amount of unique numbers will show up in the set without calculating each combination? 30 and 15 are so...symmetric to 45 that I have to imagine there's some easier way to do this. I know that doing this for combinations of 2 sets of three numbers from 1-10 looks easy (and it is), but I'm hoping that by looking at this, there's a more general formula for which you can determine the amount of uniques with combinations of p sets of q parts up to r. For instance, how many different combinations of 50 sets of 5 numbers ranging from 1-1000 will have exactly 200 unique numbers?

I hope someone can either help me out or point me in the right direction. Many thanks.

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