Triangles are drawn using the corners of a 10-sided polygon. How many of them share no side with the polygon?

All triangles: ${}^{10}{C}_3=120$. Sharing exactly 2 sides (3 consecutive corners): 10. Sharing exactly 1 side: each of 10 sides with a third corner not next to it, $10\times 6=60$. No common side: $120-10-60=50$.