How many 1st cousins 6x removed?

The famous abolitionist martyr John Brown
  http://fabpedigree.com/s032/f250617.htm
was my 1st cousin 6 times removed.

I mentioned this to someone the other day and they
laughed, thinking this was a distant relation****p.
About how many 1st cousins 6x removed does
an average person have?  How does this compare with
the number of 6th cousins 1x removed?

I don't expect an exact answer -- just a "back of
envelope" estimate.  There is an interesting little
"gotcha" in the calculation.

Other interesting puzzles arise when one tries to
apply simple math to genealogy.  For example,
I'm a descendant of Charlemagne, who was born
1266 years ago.  Any idea how many distinct
lines of descent, known or unknown, I (or you) have
from Charlemagne?

James Dow Allen