On a sheet of paper, there are 10 points drawn, 8 of which are blue and 2 are red. The two red points are connected to all the blue points, but not to each other. All the blue points are connected to each other. When two points are connected, this connection is made by a single segment. In how many different ways can we start from one red point and reach the other, passing through each point at most once?
Solution

Let's call the two red points $A$ and $B$. Starting from $A$, passing through exactly one blue point, there are 8 possibilities; passing through two blue points, there are $8 \cdot 7=$ 56 possibilities; passing through three blue points, there are $8 \cdot 7 \cdot 6=336$ possibilities. Following this reasoning, we see that the number of possibilities of going from $A$ to $B$ is $8+8 \cdot 7+8 \cdot 7 \cdot 6+\ldots+8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=109,600$. Of course, from $B$ to $A$ the result is the same, that is, we can connect the two red points in $2 \cdot 109,600=219,200$ different ways.