It is said (citation needed) that most if not all Europeans are descendents from Charlemagne in some way. How true is that? Is it even plausable statistically and mathematically? How many generations do we have to go back to find the ancestor that makes all of us (define?) remote cousins?
Greetings cousin! According to my own questionable-but-nevertheless-interesting family genealogy info, Charlemagne is my 34x-Great-Grandfather, so on the unlikely assumption that he's our closest common relative, I'll leave you to calculate what nth cousins that makes us, and how many other 34x-Great-Grandchildren he might have :)
Questions on the human genome and inter-relatedness are more up the alley of geneticists (/r/AskScience Biology category) and anthropologists (/r/AskAnthropology), so this is the thread you really want ...
... but are common here too, most often in the context of Charlemagne, Genghis Khan, or William the Conqueror. Here are the most recent posts on this:
and see this one, which has a longer explanation of the math involved, plus links to several other posts, including that AskScience AMA above
A couple of caveats when reading these threads, since comments are not always clear about what they are referring to:
"Descendants" are a given person's children's children's children's ... children. As a hypothetical example, imagine that after 2 generations he has 4 children (2 sons and 2 daughters), each of whom have 4 children (2 sons and 2 daughters): that's 4 + (4*4) = 20 descendants.
"Male-line descendants" are his sons' sons' sons' ... sons only. Using the above example, after 2 generations he has 2 + (2*2) = 6 male-line descendants (that is, his two sons and their sons; his daughters, his sons' daughters, and his daughters' children are excluded). This is the group that Y-chromosome DNA evidence refers to (see next point).
Y-chromosome DNA evidence does not prove descent from a him personally, but from people with genetic markers which he is assumed to have shared
edit: formatting