A perplexing math problem

[GW2500]

So I was hoping someone could figure out the error in my math, b/c right now I'm confused. So this is what I was thinking about, its regard to generations.
So a person was created from two people (mom and dad). That person's parents have a mom and dad, and those parents had moms and dads. Basically to figure out the number of great grandparents a person has its an exponential equation. 2^x (w/ x being number of generations) will give you how many direct descendants you have from a certain number of generations. So great grandparents are 3 generations ago, so 2^3 gets you 8, which is correct. Here's where it gets tricky. Go back 2000 years ago. So lets say the average generation is 30 years (which I imagine its even less than that). So 2000 years divided by 30 would give you 66 and 2/3. So 66 generations. Now apply the formula 2^66 . Its a really large number, well past trillions, and given that the population is the largest its ever been at 6.9 billion obviously there is a flaw in my thought. The thing is I can't figure out where I am wrong. Any one out there know what I'm missing?

 

[Ron Newman]

As you go back more and more generations, you're counting some of the same people multiple times as the occupy more than one position in your ancestry tree. For instance, your mom's great-great-great-great-grandfather might also be your dad's great-great-great-great-grandfather.

If we assume for a moment that all humans ultimately descend from a hypothetical man named Adam and a woman named Eve, then if you go back far enough they will be repeated in every 'column' of your tree.

 

[GW2500]

Ah, very true and smart of you. Thanks