Was ist korrekt: damit oder ? 1) die Jugendlichen gehen am Wochenende in die Diskothek, um zu tanzen und zu hören

The correct word to use in this case is "damit". Now let"s explain why "damit" is the correct choice for each sentence:

1) "Die Jugendlichen gehen am Wochenende in die Diskothek, um zu tanzen und zu hören." - In this sentence, the conjunction "um zu" is used to express the purpose or intention of an action. It is followed by an infinitive verb form ("tanzen" and "hören"). The conjunction "damit" is used to express the purpose or intention in a more concise way, so the correct sentence would be: "Die Jugendlichen gehen am Wochenende in die Diskothek, damit sie tanzen und hören können."

2) "Im Jugendklub gibt es viele Möglichkeiten für die jungen Leute, ihre Freizeit sinnvoll zu verbringen." - In this sentence, the phrase "für die jungen Leute" indicates that the youth club provides opportunities for the young people. The verb phrase "ihre Freizeit sinnvoll zu verbringen" expresses the purpose or intention of how they can spend their free time. To express this purpose or intention more concisely, we can use the conjunction "damit". So the correct sentence would be: "Im Jugendklub gibt es viele Möglichkeiten für die jungen Leute, damit sie ihre Freizeit sinnvoll verbringen können."

3) "Einige gehen ins Sportzentrum oder _______" - In this sentence, we need to complete the sentence with the correct conjunction. The missing word should express the purpose or intention of going to the sports center. The correct word to use is "damit": "Einige gehen ins Sportzentrum, damit sie Sport treiben können."

In summary, "damit" is the correct word choice in all three sentences because it succinctly expresses the purpose or intention of the actions mentioned. Using "damit" helps to make the sentences more concise and clear.