Hints on a problem/is my solution correct?

There are 15 different types of items and n number of poeple each have 6 of them such that none of the n people have the sam set of items. Everybody receives another item from the group of 15. Does there exist a way to give it to all n people such that they each still have a different set of 7 items?

First off, does it not matter at all what n is? Im assuming n just needs to be small enough so that the criteria that none of the n people have the same set of items fits??

Also, is it enough to prove that 15C7 > 15C6? Because there are more combinations with 7 items, for every unique set of 6 items there will be a unique set of 7 items since there are more combinations? (i mean this seems super trivial if its the case...)

Im just kinda confused because its suppose to be a graph thoery/set problem, but it seems like this can be done with basic counting?

Can someone give me some hints on how to start this question/if my reasoning so far is corret (no solutions please).

thanks