15. (4b, 8-11) In Anchuria, a checkers championship is being held in several rounds. The days and cities for the rounds are determined by a draw. According to the championship rules, no two rounds can take place in the same city, and no two rounds can take place on the same day. Among the fans, a lottery is organized: the main prize goes to the person who correctly guesses before the championship begins in which cities and on which days all the rounds will take place. If no one guesses correctly, the main prize will go to the championship organizing committee. There are eight cities in Anchuria, and the championship is allocated a total of eight days. How many rounds should the organizing committee schedule to have the highest probability of winning the main prize themselves?
# Solution.

In an $8 \times 8$ tour table, you need to select $k$ cells such that no more than one cell is chosen in any row or column. The value of $k$ should be chosen to maximize the number of combinations.

The number of combinations is given by $C_{8}^{k} A_{8}^{k}=\frac{8!\cdot 8!}{(8-k)!\cdot(8-k)!\cdot k!}$, where $C_{8}^{k}$ is the number of ways to choose $k$ cities from eight without regard to order, and $A_{8}^{k}$ is the number of ways to choose $k$ days from eight with regard to order, since each day must correspond to one of the cities. From the table

| $k$ | Number of Combinations |
| :---: | :---: |
| 0 | 1 |
| 1 | 64 |
| 2 | 1568 |
| 3 | 18816 |
| 4 | 117600 |
| 5 | 376320 |
| $\mathbf{6}$ | $\mathbf{5 6 4 4 8 0}$ |
| 7 | 322560 |
| 8 | 40320 |

it is clear that the maximum number of combinations occurs when $k=6$.

Answer: 6.