10.5. Five athletes came to training with their balls, and when leaving, each took someone else's. In how many ways is this possible.
Answer: 44.

Solution. First, let's assume that no two athletes have exchanged balls. Imagine them sitting around a round table. Then, depending on their different relative positions, they would choose the ball of the next (for example, clockwise) athlete. However, this is only possible if 5 are seated at one table, which can be done in $4!=24$ ways.

Now consider the case where two athletes have exchanged balls. These two out of five can be chosen in 10 ways, and the remaining three can be seated around the round table as in the first case, which can be done in $2!=2$ ways. Thus, in this case, we get 20 ways.

Since only one pair of athletes can exchange balls, the total number of ways will be $24+20=44$.

Comment. Consider only one of the cases -3 points.