So, in this entry about finding the formula to determine the total number of possible relationship configurations for any group of n people, blaisepascal observed that the sum we came up with did not consider the case where everyone is involved with everyone else, which is something we had intended to include. The previous equation, therefore, has an off-by-one error. The correct form is:
This is, as many people have observed, essentially the standard “pick r of n permutations” equation, which (had we been thinking along those lines) we likely should’ve recognized from the start. And, to be fair, one of the more math-geeky among us said something like that early on, but it took much scribbling on many sheets of paper to prove it.