I am having a hard time visualizing and conceptualizing what a k-wise intersection is.
I am guessing 3-wise intersection for 3 sets: $S_1,S_2,S_3$ would be $(S_1 {\cap}S_2{\cap}S_3)$ and 2-wise intersection for 2 sets $(S_1{\cap}S_2) + (S_2{\cap}S_3) + (S_2{\cap}S_3)$ and so on.
But if we have more sets, say if we have 10 sets, how can I conceptualize it in my head?
Would the number of k-wise intersection just be representing $\binom{n}{k}$ where n is the number of sets and k is the k-wise intersection?
Is this correct way to think about k-wise intersection?
Everything which looks like $$ S_{i_1}\cap S_{i_2}\cap \dots \cap S_{i_k} $$ is a $k$-wise intersection. If you have $n$ sets $S_1,S_2,\dots,S_n$, then the number of possible $k$-wise intersections of these sets is indeed $\binom{n}k$. Every size $k$ subset $A=\{i_1,i_2,\dots,i_k\}$ of $\{1,2,\dots,n\}$ corresponds to exactly one $k$-wise intersection, namely the one above.
When using the Principle of Inclusion Exclusion, you will have to sum over all possible $k$-wise intersections, for each $k$. This summation is either written as $$ \sum_{|A|=k}N({\textstyle\bigcap_{i\in A}}S_i) \hspace{1cm}\text{or}\hspace{1cm}\sum_{1\le i_1<i_2<\dots<i_k\le n}N(S_{i_1}\cap S_{i_2}\cap \dots \cap S_{i_k}) $$