Spot It

Was pretty impressed with this game (Spot-It) today at the Velzen family Christmas.  I didn't really enjoy playing it, because my eyes and brain don't think quick enough to spot patterns and matches (I despise "speed" games and prefer slower-paced games) but it was a great math problem!!

The basic idea is that every card has 8 items on it, and you are supposed to find what matches from cards to cards.  The thing that impressed me was there was about 50 different pictures on the various cards, but every card shares one and only one picture every other card.  That is, take any pair of cards and they will have exactly one thing in common.

Can you do this with cards with 9, 10, 11 items on it?  How many different types of pictures do you need for 8 items per card so you don't ever have pairs of cards with 2 matches?  How many different cards are possible for x number of objects and y number of objects per card?  And how in the world do you begin to list all the possibilities in a useful, organized manner.  Would sure love to see a graph of this problem -- is it possible to draw in 2 dimensions?  Is it an 8-dimensional graph/problem?