6. Every week, listeners choose the ten most popular songs via SMS. It is known that 1) the same set of songs in the same order is never chosen twice in a row; 2) a song that has once dropped in the ranking will not rise again in the future. What is the maximum number of weeks that the same 10 songs can remain in the ranking?
Answer: 46 weeks.

Sketch of the solution. Suppose that for two consecutive weeks, the same set of 10 songs was on the list. Since the order of the songs is different, at least one song in the list has moved up in the ranking, and at least one has moved down. Since a song, once it has moved down, does not move up in the ranking, let's count the total number of possible rises. The last song in the initial list could rise no more than 9 times, the second-to-last no more than 8 times, and so on. Thus, there are no more than $9+8+\ldots+2+1=45$ rises. Therefore, the list can stay in the ranking for no more than 46 weeks.

It remains to provide the necessary example. First, the song that was in first place drops 9 weeks to the last place, then the second song drops 8 weeks. The third song in the initial list drops 7 weeks, and so on.