Question

How many subgraphs with at least one vertex does K3 have?

Answer 1

I think it should be $17$ (with at least one vertex)

And asked about  exactly with $3$ vertices then $2^{n(n-1)/2} =8$

But here  at least  $1$ vertex so

• $3$ subgraph as consider single - vertices ( question does not mention about distinct so we will consider all cases )
• $3$ subgraph consider $2$ vertices 
• $1$ subgraph consider $3$ vertices
• $3$ subgraph consider  $2$ vertices, one edge
• $3$ subgraph consider $3$ vertices, $2$ edges
• $3$ subgraph consider $3$ vertices, $1$ edge
• $1$ subgraph consider $3$ vertices, $3$ edges

So, total $17$

Comments

Well, yes. But since the question doesn't say that the vertices are labelled, you just assume that there are no labels, and then the answer will be $7$ as shown by Arayana.

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haa ,arayana  is right .. but in question , does not mention they are distinct ...so i consider as label and this question is from rosen and i think ans given is 17 ... so i assumed ...

anyway thanks for edit ..now it looks good ...

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$\sum_{r=1}^{3}\binom{3}{r}2^{\frac{r(r-1)}{2}}$

Answer 2

Therefore there are 7 subgraphs possible in case of unlabeled vertex in k3 having atleast one vertex.