I have a set of $X$ symbols which I have to put in cards of $N$ (where $X > N$) slots such that every card must have exactly $1$ symbol in common with each other. What I want to know is, how many cards can I form? I don't even know how to mathematically formulate this, how do I go about this?
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Do you mean every set of two cards should have exactly one symbol in common? – 雨が好きな人 May 02 '19 at 14:17

I meant that every card must have a common symbol with every other card. Like, suppose we have a card with symbols "A B C" all other cards must have one of those symbols – Alexey May 02 '19 at 14:21

1Well, you need to say more than that. If my symbols were, as you say, $A,B,C$ then I could have infinitely many cards each just using the symbol $A$. That satisfies your stated requirements, but I seriously doubt you would allow that configuration. – lulu May 02 '19 at 14:40

I would assume the cards have to be distinct. – 雨が好きな人 May 02 '19 at 14:42

2"Every card must have **exactly** 1 symbol in common..." It sounds like you are referring to the game "Spot It." See the following: [What is the math behind the game Spot It?](https://math.stackexchange.com/questions/36798/whatisthemathbehindthegamespotit?rq=1), [What are the mathematical computational principles behind this game?](https://stackoverflow.com/questions/6240113/whatarethemathematicalcomputationalprinciplesbehindthisgame/), [Algorithm to Create all Spot It cards](https://math.stackexchange.com/questions/500336/algorithmtocreateallspotitcards). – JMoravitz May 02 '19 at 14:57

@JMoravitz exactly what i was looking for, thank you! – Alexey May 02 '19 at 16:16

See also [this answer](https://math.stackexchange.com/a/3030747/177399) for discussion of how your problem relates to Steiner systems. – Mike Earnest May 02 '19 at 16:29