We've all got yichus

A commentator on Yeranen Yaakov's post on Rashi writes,

- Not few people can trace their ancestors up to Rashi, which isn't that amazing, as he was famous ever since his lifetime, and with only some 20,000 Jews in Europe at the time, probably all Ashkenazim have several lines to Rashi. Of course, as he didn't have sons, nobody is ben achar ben.
- Rashi's ancestry goes back to Dovid hamelech.

The truth is that, by the same logic, probably every single Ashkenazi Jew is descended from (at the very least) one famous Ashkenazi rabbi or another, if only you go back several hundred years. And kal vachomer, we're all descended from King David.
You only have to look at the equivalent stats for Europeans in general. In a fascinating article in the Atlantic a few years ago, Steve Olson wrote (based on the research of Joseph Chang, a statistician at Yale University):

The mathematics of our ancestry is exceedingly complex, because the number of our ancestors increases exponentially, not linearly. These numbers are manageable in the first few generations -- two parents, four grandparents, eight great-grandparents, sixteen great-great- grandparents -- but they quickly spiral out of control. Go back forty generations, or about a thousand years, and each of us theoretically has more than a trillion direct ancestors -- a figure that far exceeds the total number of human beings who have ever lived....
20 percent of the adult Europeans alive in 1000 would turn out to be the ancestors of no one living today (that is, they had no children or all their descendants eventually died childless); each of the remaining 80 percent would turn out to be a direct ancestor of every European living today....
[A]lmost everyone in the New World must be descended from English royalty -- even people of predominantly African or Native American ancestry, because of the long history of intermarriage in the Americas. Similarly, everyone of European ancestry must descend from Muhammad. The line of descent for which records exist is through the daughter of the emir of Seville, who is reported to have converted from Islam to Catholicism in about 1200. But many other, unrecorded descents must also exist.

The question, therefore, isn't which of us is descended from King David -- but only which of us can actively trace it. (And I'm not just saying that because I can only trace back Polish peasants.) As a side point, Olson says that the most recent common ancestor of all Europeans lived about 600 years ago; I'd love to know when the most recent common ancestor of all Jews, or all Ashkenazi Jews, was.
Also following on from the Atlantic article (as Olson explicitely states) is that if you wait enough generations, in time each and everyone one of us will be a direct ancestor of all of humanity. In the same vein, and probably a lot quicker, wait enough generations and each and every one of us will be an ancestor of all the Jews in the world.
Nice thought.