Graph Count

You are given an undirected graph with $N$ nodes and $M$ edges. There are exactly $K = \frac{N(N-1)}{2} - M$ unordered pairs $(i, j)$ such that the edge $(i, j)$ is not present in the graph. For each of these pairs, you can choose whether to add the edge $(i, j)$ to the graph. Out of all $2^K$ ways, find the number of ways in which each vertex has degree $2$ in the final graph. Since the answer c