I'm having difficulty finding the number of subgraphs of a triangle... the solutions of the textbook I'm using says the answer is $18$, but I am counting $17$ in the following way:
$$\binom{3}{1}\cdot 2^{\binom{0}{2}} + \binom{3}{2}\cdot 2^\binom{2}{2} + \binom{3}{3}\cdot 2^\binom{3}{2} = 17\,,$$
since there are $\binom{0}{2}=0$ edges total one can place in a graph with $1$ vertex, etc. I am not sure if I am using the right method to count, since I am counting one less than what the answer is supposed to be. Any help would be appreciated!