A music streaming service proposes songs classified in $10$ musical genres, so that each song belong to one and only one gender. The songs are played one after the other: the first $17$ are chosen by the user, but starting from the eighteenth the service automatically determines which song to play. Elisabetta has noticed that, if one makes the classification of which genres they appear several times during the last $17$ songs played, the new song always belongs to the genre at the top of the ranking or, in case of same merit, at one of the first genres.
Prove that, however, the first $17$ tracks are chosen, from a certain point onwards the songs proposed are
all of the same kind.
1. **Initial Setup and Definitions**:
   - There are $10$ musical genres.
   - The first $17$ songs are chosen by the user.
   - From the $18$th song onwards, the service automatically determines the genre based on the last $17$ songs played.
   - The new song always belongs to the genre that appears most frequently in the last $17$ songs. In case of a tie, it belongs to one of the tied genres.

2. **Observation on Majority and Minority**:
   - If a genre is in the sole majority (i.e., it appears more than any other genre in the last $17$ songs), then the next song will be of that genre.
   - If a genre is in the minority (i.e., it appears less frequently than some other genres), it will continue to remain in the minority as long as the majority genre(s) continue to dominate.

3. **Analysis of the Number of Songs and Genres**:
   - Since $17$ is not divisible by $10$, it is impossible for all genres to appear an equal number of times in the last $17$ songs. Therefore, there will always be at least one genre that appears more frequently than others (i.e., a majority genre).

4. **Behavior of the System Over Time**:
   - Consider the sequence of songs starting from song $i$ to song $i+16$ (a total of $17$ songs).
   - If a genre is in the minority at song $i$, it will not be chosen for song $i+17$ because the service will choose a genre from the majority.
   - As the process continues, the majority genre will keep getting reinforced because it will be chosen more frequently, thereby increasing its count in the last $17$ songs.

5. **Convergence to a Single Genre**:
   - Over time, the majority genre will dominate the sequence of songs.
   - Once a genre becomes the sole majority, it will continue to be chosen for subsequent songs.
   - This leads to a situation where, eventually, all songs proposed by the service will be of the same genre.

6. **Conclusion**:
   - Regardless of the initial $17$ songs chosen by the user, the system will eventually converge to playing songs of a single genre.

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