Since a pentagon has 5 vertices, in order to compute how many pentagons in an octagon, which, instead, has eight vertices, we can compute the number of ways in which we can take 5 vertices out of 8. Computing the number of pentagons that can be inscribed in an octagon amounts to compute in how many ways we can choose 5 points out of 8, i.e. \(8C5 = \frac{8!}{5!3!}\). The same holds for the number of triangles, i.e. they can be computed as \(8C3 = \frac{8!}{3!5!}\). The two quantities are then equal and answer is C!.