The game of Dobble (will edit in a link later) involves a set of bespoke playing cards covered in symbols or small pictures - a dog, an arrow, a pencil, a tree etc.
Each card contains eight such symbols, and any two cards will always have exactly one symbol in common.
There are multiple variations of how the game is played but the basic object is to be the first player to spot the common symbol between two cards (typically between one in your hand and one on the table) and shout out the appropriate name to claim the point.
There are 8 symbols on each card.
How many different symbols must there be in total?
How many cards must there be in the full set?
Is there a general formula for the above as the number of symbols on each card changes?
Assume (to avoid degenerate answers) that:
each symbol appears on multiple cards;
every possible pair of symbols occurs on exactly one card.