I have deleted what was here as it was based on a misunderstanding of the problem. My current understanding of the problem is that, given positive integers $k\le n$ one wants the largest number of $k$-element subsets of an $n$-element subset, no two intersecting in more than one element. This has indeed been studied as part of coding theory. In the language of coding theory, we want the biggest binary code of length $n$, all codewords being of weight $k$, the code having minimal distance $2k-2$. The case $k=3$ is discussed at http://www.research.att.com/~njas/sequences/A001839 and the notes there also give references to the general case.

I have deleted what was here as it was based on a misunderstanding of the problem. My current understanding of the problem is that, given positive integers $k\le n$ one wants the largest number of $k$-element subsets of an $n$-element subset, no two intersecting in more than one element. This has indeed been studied as part of coding theory. In the language of coding theory, we want the biggest binary code of length $n$, all codewords being of weight $k$, the code having minimal distance $2k-2$. The case $k=3$ is discussed at http://oeis.org/A001839 and the notes there also give references to the general case.

I think what's wanted here is the number of ways of partitioning a set of size $n$ into subsets of size $r$ (presumably with some elements left over if $n$ is not a multiple of $r$). This isn't really an MO-type question, but for the specific example of partitioning a set of size 9 into three subsets, each of size 3, the answer should be
(9! / 3! 6!)(6! / 3! 3!)(3! / 3! 0!) all divided by 3!.

I have deleted what was here as it was based on a misunderstanding of the problem. My current understanding of the problem is that, given positive integers $k\le n$ one wants the largest number of $k$-element subsets of an $n$-element subset, no two intersecting in more than one element. This has indeed been studied as part of coding theory. In the language of coding theory, we want the biggest binary code of length $n$, all codewords being of weight $k$, the code having minimal distance $2k-2$. The case $k=3$ is discussed at http://www.research.att.com/~njas/sequences/A001839 and the notes there also give references to the general case.