The answer is 10. More specifically what it takes to get all the possibilities is find a number n that is so that 2^n it bigger or equal to the number of vials

If you have 4 vials for example (a, b, c and d), you’d need 2 people (A and B). Here are the possibilities for each vials if A is injected with b and c, and B is injected with c and d.

a: no one dies

b: A dies

c: AB dies

d: C dies

let’s do 8 vials now (a, b, c, d, e, f, g, h, i). You’d need 3 people (A, B, C). A has bcdh, B has cef and C has dfgh

a: no one dies

b: A dies

c: AB dies

d: AC dies

e: B dies

f: BC dies

g: C dies

h: ABC dies

Now if we have 4 people A, B, C and D, we have:

a: no one dies

b: A dies

c: AB dies

d: AC dies

e: AD dies

f: ABC dies

g: ABD dies

h: ACD dies

i: B dies

j: BC dies

k: BD dies

l: BCD dies

m: C dies

n: CD dies

o: D dies

p: ABCD dies

So the pattern is (left is n person, right is n vials)

1: 2

2: 4

3: 8

4: 16

This follows a 2^n rule, considering n is the number of people

So we need to find the number n that makes 2^n smaller or equal to 1000

2⁹ is 512, then you have 2¹⁰

So you need 10 people.

My arms hurt typing this lol.

edit: god ther formating