If anyone can offer any guideline on how to start this type of problem, I'd be really thankful. It's a home-work problem, so please just offer some hints not entire solutions. Thank you.
Let L be a set of n elements. Count the number of ordered pairs (A,B) of subsets such that $ \varnothing \subseteq A\subseteq B \subseteq L$. Let c(j,k) denote the number of such ordered pairs for which A contains j elements and B contains k elements. Show that: $\sum_{}^{0\leqslant j\leqslant k\leqslant n} c(j,k) \cdot x^j \cdot y^k = (1 + y + yx)^n$
The left side is just the generating function for $j$ elements of $A$ and $k$ elements of $B$. We'd like to come up with a combinatorial explanation for the right hand side. Notice that $x$ is attached to $A$ and $y$ is attached to $B$.
There are $n$ elements. Each of these elements are in neither $A$ nor $B$ (corresponding to $1$), are in $B$ but not in $A$ (corresponding to $y$), or are in both $A$ and $B$ (corresponding to $xy$). Since there are $n$ elements and you're just counting them, these possibilities are given by $(1 + y + yx)^n$.
Your goal is to understand this and make it rigorous (or at least as rigorous as you think is necessary to connote understanding).