Number of sub graphs of a complete graph

Question

Let $G$ be a complete graph with $m$ edges and $n$ vertices, and $P(G)$ be the set of all possible sub graphs of $G$. Then the number of elements in $P(G)$, i.e., $|P(G)|=2^n+\binom m1+\binom m2+...+\binom mm.$ I believe that this formula is true. Your valuable comments are most welcome, please.

Answer 1

As noted in the comments, a correct formula seems to be $$\sum_{k=0}^na_k{\binom n k}$$ where $a_k=2^{\binom k 2}=2^{k(k-1)/2}$.