How to determine the number of directed/undirected graphs? I'm kind of stuck on this homework problem, could anyone give me a springboard for it? If we have $n\in\mathbb{Z}^+$, and we let the set of vertices $V$ be a set of size $n$, how can we determine the number of directed graphs/undirected graphs/graphs with loops etc.? Is there a formula for this? I feel like it can be done using combinatorics but I can't quite figure it out. Any ideas?
Thanks!

A start: We will show how to count labelled, loopless, undirected graphs. There are $\binom{n}{2}$ ways to choose a set $\{u,v\}$ of two vertices. For every such set, we say yes or no depending on whether we have decided to join $u$ and $v$ by an edge.  Alternately, but somewhat less concretely, let $P$ be the set of all (unordered) pairs. This set $P$ has cardinality $\binom{n}{2}$. To specify a loopless undirected graph, we choose a subset of $P$ and connect any unordered pair in that subset by an edge. How many subsets does $P$ have?
To extend to graphs with possible loops (but at most one per vertex) there is a similar yes/no process. 
