Draw all possible graphs that can be constructed from the vertices $V_1=\{a,b,c\}$. Answer: Pic: I believe this is all of them (bottom three are all the same).
How many such graphs have no edges? Answer: $1$.
How many such graphs have exactly one edge? Answer: $3$.
How many such graphs have exactly two edges? Answer: $3$.
How many such graphs have all three edges? Answer: $1$.
How many total graphs are there? Answer: $8$.
Now consider a vertex set of size $4$, $V_2 = \{a, b, c, d\}$.
How many possible edges exist over $V_2$? Answer: $C(4,2) = 6$.
How many unique graphs can be constructed from $V_2$? Hint: In any such graph, each edge is either present or absent. Answer: 4 2-way choices 2^4 = 16
How many unique graphs can be constructed from $V_2$ with exactly three edges? Hint: One must choose where to place the three edges.
Now generalize these results for a vertex set of size $n$, i.e., $V_3 = \{a, b, c,\dots,a_n\}$ where $|V_3| = n$.
How many possible edges exist over $V_3$?
How many unique graphs can be constructed from $V_3$? Justify your answer.
How many unique graphs can be constructed from $V_3$ using exactly $k$ edges? Justify your answer.
Please let me know if my answers for the first few are correct and any help is appreciated with the rest!! Thanks!!