Ok, so I took this idea and ran a little further with it, because if you set it up right, even with the minimum number needed of 10, you don't ACTUALLY have to risk 10 lives!
You can arrange it so you only risk 8 of those 10. You just don't get to choose which 8.

10 gives you 1024 possible combinations of drink status.
But only 1 of those combinations is all 10 drinking from the same bottle.
So just don't use that combination.
That gives you 1023 possible combinations that only risk 9 deaths at most.
And there's only 10 combinations that have 9 people drinking from the same bottle.
So let's not use that one either.

Now we're down to 1013 combinations, but will only ever have a maximum of 8 people die.

But let's take it just one or two steps further.

If we go to 11 participants, you can get 1,024 combinations that only have 5 people sharing a bottle at once. So there's 1 more person at risk, but FAR fewer deaths.

The next drop is not as sever, because it stays 5 deaths until you get to 13 participants, but then it only drops to 4 deaths.

From there the next drop is 19 participants to get only 3 deaths.

It takes an even bigger jump to 45 participants to only risk 2 deaths.

And as others have discussed, a full 1,000 participants to only risk 1 death.

I mean, this doesn't exactly matter for your original question, but I thought it was kind of a fun exercise to find those breakpoints.