Each vertex of a nine-sided polygon is connected to the other vertices with line segments. How many of these line segments lie inside the polygon?
+310 Here are two solutions.
1) It takes 2 vertices to make a line, so there are \({9 \choose 2}\) line segments. We have to subtract 9 from this because the 9 sides don't count.
\({9 \choose 2}-9=27\)
2) From each vertice, there are 6 other vertices to draw a diagonal from (we can't include the vertice itself or its two adjacent vertices). In all, there are \(\frac{9*6}{2}=27\). We divide by 2 because we count each diagonal twice (once from each endpoint).
