Choosing dishes and sharing...

Hi,

Here is a problem I'm having a hard time wrapping my head around.

10 people go to a restaurant to order 10 dishes out of 50 distint dishes. If people can share the dishes among themselves then how many distinct eating experiences are possible?

First of, there are 2 cases for whether repetition is allowed of not. In the first case, if repetition is not allowed then there are ^{50}P_{10} distinct experiences, else it is 50^{10}. (Permutation because the order of dishes matters in creating distinct experiences.)

How to consider sharing of the 10 dishes among the 10 people? In the no repetition case, 10 distinct dishes can be shared among 10 people in 10^{10} ways, so the total is ^{50}P_{10 }X 10^{10 }? What about the case where when repetition allowed, i.e., all people can have the same dish?

Any help is appreciated.

Thanks

Re: Choosing dishes and sharing...

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.

Re: Choosing dishes and sharing...

Thanks for the quick response. Your interpretation makes sense (Sharing is equivalent to choosing since order doesn't matter anymore). I was thinking that sharing should mean more distinct experiences.

So in summary, (please correct me if I am wrong)

No repetition: 50P10 if no sharing and 50C10 if there is sharing.

With repetition: 50^10 if no sharing and (50C10 + 50C9 + 50C8 + ...+ 50C1) if there is sharing

Thanks