What is the largest $k∈N$ with the property that there are non-trivial events $A_1,...,A_k ⊆ Ω$ such that $A_1,...,A_k$ are independent?

Question

Let $\mathbb{P}$ be the uniform distribution on $Ω = {1,2, . . . ,10}$. An event $A⊆Ω$ is said to be non-trivial if $0<P(A)<1$. What is the largest $k∈N$ with the property that there are non-trivial events $A_1,...,A_k ⊆ Ω$ such that $A_1,...,A_k$ are independent?

Found this question in the textbook but really unsure of how to approach this, any advice is appreciated.

Answer 1

Hint:

Subsets $A$ and $B$ are independent here iff: $$10|A\cap B|=|A|\times|B|$$

This (together with the fact that $A$ and $B$ are non-trivial) provides info about $(|A|,|B|)$.

Now find $2$ non-trivial independent subsets and wonder if you can find a third one.