How to solve below question on probability ?

Question

Answer 1

$\frac{1}{6} \text{ should be the correct answer.}$

Answer 2

For Triangle one Edge need to select from 3 Edges of 123 by 3c1 ways out of 9c1.

now for any k there is 3 possibilities out of 6 vertices.

so (3 * 3c1) / (6 * 9c1) = 0.1666