13. Random graph (from 8th grade. 3 points). On a plane, 20 points were pairwise connected by segments (edges) ${ }^{3}$ - each point with each other. Then, 35 edges were randomly selected and removed. Find the probability that the resulting graph is connected (i.e., from any point, you can reach any other point by edges).
Solution. A graph is disconnected if it can be divided into two groups of points, such that no point in the first group is connected by an edge to any point in the second group. Suppose the first group has $n$ points, where $2 \leq n \leq 10$, and the second group has the remaining $20-n$ points. The remaining edges after removal connect points within the first group and points within the second group, but there are no edges connecting points from one group to the other. From each of the $n$ points in the first group, there were $20-n$ edges to points in the second group, and all these edges have been removed. Therefore, at least $n(20-n)=20n-n^2$ edges have been removed. For $2 \leq n \leq 10$, the minimum value of this expression is

$$
r(2)=40-4=36
$$

Thus, it is impossible to remove 35 edges such that each group retains at least two points. Therefore, the graph is disconnected only if there is a single point that is not connected to any other (an isolated point).

Let $A$ be some point. Out of the 19 edges connecting it to other points, all 19 have been removed. From the remaining $\frac{19 \cdot 18}{2}=171$ edges, 16 more have been removed. This can be done in $C_{19}^{19} \cdot C_{171}^{16}=C_{171}^{16}$ ways. Initially, the graph had $C_{20}^{2}=190$ edges. Therefore, the 35 edges can be removed in $C_{190}^{35}$ ways. Thus, the probability that point $A$ is isolated is $\frac{C_{171}^{16}}{C_{190}^{35}}$.

It remains to account for the fact that any point can be the isolated one, and there are 20 mutually exclusive and equally likely cases. Therefore, the probability that the graph is disconnected is $\frac{20 C_{171}^{16}}{C_{190}^{35}}$.

Answer: $1-\frac{20 C_{171}^{16}}{C_{190}^{35}}$.