27.7. Ten points are marked on a circle. How many non-closed, non-self-intersecting nine-segment broken lines with vertices at these points exist?
27.7. The first point can be chosen in ten ways. Each of the following eight points can be chosen in two ways, as it must be adjacent to one of the previously chosen points (otherwise, a self-intersecting broken line would result). Since the beginning and end are not distinguished in this calculation, the result must be divided by 2. Therefore, there are $10 \cdot 2^{8} / 2=1280$ broken lines.