Grouping Elements

[kaleidoscope]

How many groups size m can you make from n elements (m<n) such that each element is used the same number of times t (t>0)?

For instance, if you have 8 teams and group them in triplets, how many triplets do you need so that each team plays the same number of times?

[ideasrule]

How many groups size m can you make from n elements (m<n) such that each element is used the same number of times t (t>0)?


Why wouldn't this be infinite? If repetition is allowed, you can make n groups, each containing m repetitions of the nth element. You can also make 2n groups (so 2 groups contain repetitions of the same element), or 3n, or 4n...


For instance, if you have 8 teams and group them in triplets, how many triplets do you need so that each team plays the same number of times?

Would (1,1,1),(2,2,2),(3,3,3)...(8,8,8) be valid? Of course you can switch the numbers around; each team just needs to play 3 times.