system
Given the following post, respond as if you're commenting on Reddit:
human
Is there a finite number of combinations possible in the human genome?  If so, how long would it take before we start "repeating" people?

I was just wondering this the other day, since each person's genome is just a sequence of base-pairs, if there is a finite number of combinations eventually it should be possible that two people (not necessarily alive at the same time) have an identical genome.  Of course, this would ignore identical twins.

I have an idea for how to figure this out, but I need some help on the specifics.  We now how many base-pairs there are in our genome (just over 3 billion), so we could use that to determine how many possible combinations there are (if it's not infinite).  We also have approximate population data/predictions that goes a long way back/forward, which can be used to find cumulative population.  As soon as the cumulative population is greater than the number of possible genome combinations, we can be sure that someone has been "repeated".

What I'm thinking is that our DNA changes so often that there will always be more combinations possible.  I'm not sure if that's how it works or not, though.
gpt
Crash course on combinatorics - even a few 'bits' can result in MANY combinations. 

For example, encode a genome in binary, 0011010101 or something like that (it's actually a higher base than 2, but just take base 2 for now). With just 10 bits, you'd have 2^10 = 1024 combinations. With a thousand bits, you'd have 1e301 combinations. That is 1 with THREE HUNDRED ZEROS AFTER IT. As a point of reference, the milky way in meters is only about 1 with 16 zeros after it (10,000,000,000,000,000 m). 

So, yes, DNA is finite, but no, you'll never ever get two of the same people. Effectively EVER.