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Human genetic code has maybe 0.2% variation in base pairs, the average individual gene has maybe 13 thousand base pairs, and the whole code has about 3.1 billion base pairs.

This is crappy, quick and dirty math, but this might be one of those high-complexity situations (like weather modeling) where minimal detail can be closer to the truth than an intermediate amount anyway. So, we'll model it as a random selection of 0.2% differentiated genes, meaning ~240,000 take ~500, which comes out to a number with 1556 digits. The amount needed to get a 50/50 chance of a repeat will be lower, with 777, almost 778 digits.

If you allow weird, implausible and mostly stillborn combinations it's closer to 2^~240,000^ . If you include mutations, it gets much much higher to the point of basically being undefined. It's more about how many changes is still human, and after that how big a genome a cell can handle.

A conservative estimate for the number of 'people combinations' is about 10^4,515,450.

There are 10^19 grains of sand on earth.

There are 10^50 atoms on earth.

There are 10^68 possible orderings for a deck of cards.

There are 10^80 atoms in the observable universe.

There are 10^120 possible chess games.

Humans have 3.2 billion base pairs. There likely has not and will never be two identically genetic humans, even identical twins are not perfectly identical due to mutations.

Now, obviously a lot of the genetic code needs to be similar, certain parts of our genetic code needs to be similar, so it isn't literally 3.2*10^9 factorial, but even a small fraction of that number is essentially infinite. I highly doubt there has ever been two identical humans or ever will be.

3.2 billion factorial is essentially arbitrarily large. It isn't even really useful to think of it as a number, as we couldn't even calculate what the number is. Visualize the largest quantity you can conceptualize, then square it until you get bored.

I'm no scientician, but wouldn't mutation mean that the possible number of gene combinations is constantly growing? And maybe even making previously non-viable permutations viable, which seems like it would really fuck with the math?

The number of possible combinations of genes is so large that the chance of a random duplication is vanishingly small. Even twins aren't exact copies because they'll have mutations unique to themselves.

To add on top of the other answers: On top of all potential genetic variation, you also have environmental and experiential variables that further differentiate people.

If you truly shuffle a deck of cards, you are likely to come up with a permutation that has never existed before.

The human genome is infinitely more complex than a deck of cards.

At least 12.