I was given this sentence
Сегодня мы с Джулией собираемся в поездку в Россию по программе обмена студентами.
- по программе — dative
- обмена — genitive
- студентами — instrumental
Why are these all in different cases? I understand that after по, nouns have to be in the dative case, but I don't know why exchange is put in the genitive, and students is in the instrumental.
The way I thought it worked is that everything referring to the initial thing you're talking about will be put in the same case, and since program is in the dative, I thought everything else would be dative, too.
Is there a general rule I can follow to understand when to put a word in a specific case? I'm really confused on how that works — especially in this sentence.
I would love as much information and as many tips as you can offer to help me with this.