The fact that they can share dishes implies that order is not important - if Andy chooses dish A and Bill dish B it's the same experience as Andy choosing B and Bill choosing A, because they could exchange dishes if they want (the ultimate in sharing). Similarly if Andy and Bill both choose A and Charlie chooses B, I think that's the same experience as Andy choosing A and Bill and Charlie both choosing B. So I interpret this question to mean: how many ways can 10 out of 50 dishes be chosen? If there are no repeats then that would be 50C10 - note that we use combination, not permutation, because the order of the dishes is not important. But supoose there is a repeat, so that only 9 distict dishes are selecetd by the ten people - the numbner of ways that 9 dishes can be picked is 50C9. And so on for cases where only 8 distinct dishes are picked, or 7, etc. So in total you have
50C10 + 50C9 + 50C8 + ...+ 50C1 distinct combinations.