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A recurring theme in a problem I am exploring is counting the number of subsets of sizenin [itex]Z^{d}_{3}[/itex] that have at leastmmutually cohyperplanar (dimension d-1) points.

For example, ifn=5,m=3,d=2, the question is: "How many ways are there to select 5 points from a 3x3 plane such that any 3 points are mutually collinear?" Because 5 points guarantee a complete line in [itex]Z^{2}_{3}[/itex], the answer is the same as 9 choose 5, specifically 126.

The specific case I am currently interested in, but most likely not the last, is the case withm=7,d=3. So, the question is: "How many ways are there to selectnpoints from [itex]Z^{3}_{3}[/itex] such that any 7 are mutually coplanar?", where n is a variable.

Is it too much to expect that there should be a fairly simple combinatorial answer to this? So far I have not been able to derive it, but I feel that it should exist.

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# Number of ways to select M cohyperplanar points in finite space

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