Descending from someone who became famous isn’t as uncommon as one might think. Someone claiming lineage back to Charlemagne, not an unusual claim, would very likely not be lying.
There are an estimated 40 generations between Charlemagne and the present generation. Two (people per generation) to the 40th power gives a person 1 trillion, 99 billion, 511 million, 627 thousand, 776 ancestors.
Two parents per generation
Each generation has 3 children
A more concrete example:
Calculating direct ancestors. There are ten generations between Mark and George Soule, one of the Pilgrims. Mark had two parents, his parents had two parents, and so on. So, back to Plymouth Rock, Mark has 1024 direct ancestors. (left)
Calculating direct descendants. If George Soule had three children who had three children and so on, after 10 generations, George would have 59,049 descendants.
Following this rate of expansion, after 40 generation, Charlemagne’s descendants would number
12,157,665,459,056,900,000
which is 12.157 quintillion and change. That’s a LOT of Christmas cards!
There’s a great explanation on this blog over here: http://nltaylor.net/sketchbook/archives/9
and this guy did the math: http://www.gly.uga.edu/railsback/GenealComp1.html
Social History 101 - the blog 