## Task 6 - 030916

a) On the circumference of a circle, 5 different points are distributed arbitrarily. How many line segments can be drawn that connect any two points?

b) How many line segments are determined if 10 points lie on the circumference of the circle?

c) Let the number of points be $n$. How many line segments can be drawn? (Justification!)
c) Let $P_{i}, i \in\{1, \ldots, n\}$ be the $n$ points on the circumference of a circle. Each point $P_{i}$ can be connected to $n-1$ points $P_{j}, j \in\{1, \ldots, i-1, i+1, \ldots, n\}$. Thus, the total number of possible directed point-to-point connections is $n(n-1)$.

Segments are not directed connections between two points. Therefore, a pair $\left(\overrightarrow{P_{i} P_{j}}, \overrightarrow{P_{j} P_{i}}\right)$ of oppositely directed connections between two points has only one corresponding segment, $\overline{P_{i} P_{j}}$. The total number of possible segments is thus half the total number of possible directed connections, which is $\frac{n(n-1)}{2}$.

a) From c), for $n=5$, the number of possible segments is $\frac{n(n-1)}{2}=\frac{5 \cdot 4}{2}=10$.

b) From c), for $n=10$, the number of possible segments is $\frac{n(n-1)}{2}=\frac{10 \cdot 9}{2}=45$.