## Task 4 - 150714

In the plane $\epsilon$, there are 50 different points arranged such that no line exists that contains three of these 50 points. Each of these 50 points is to be connected to every other point by a line segment.

a) Determine the number of connecting line segments!

b) Suppose the 50 points are the vertices of a convex 50-gon. Determine the number of diagonals of the 50-gon!
a) For choosing the first endpoint of one of the connecting lines, there are exactly 50 possibilities. In each of these, there are then exactly 49 possibilities to choose one of the remaining endpoints as the second endpoint of one of the connecting lines. Through these overall $50 \cdot 49=2450$ possibilities, all the desired connecting lines are obtained, and each line is obtained exactly twice (since each endpoint appears once as the first and once as the second endpoint).

Therefore, $2450: 2=1225$ is the number of all the desired connecting lines.

b) Of these 1225 connecting lines, exactly 50 form the sides of the convex 50-gon, and the remaining ones are the desired diagonals. Their number is thus $1225-50=1175$.

Solutions of the I. Round 1975 adopted from $[5]$

### 4.17.2 II. Round 1975, Class 7