80 participants in total, each can belong to any number of $\displaystyle a,b,c$ (it is not stated but I assume they can belong to none of the sets as well).

$\displaystyle |a|=50, |b|=40, |c|=30$

Either one of these two following scenarios is correct, we have to figure out which one:

(1) $\displaystyle |a\cap b|=30, |a \cap c|=20, |b \cap c|=20, |a \cap b \cap c|=10$

(2) $\displaystyle |a\cap b|=10, |a \cap c|=10, |b \cap c|=20, |a \cap b \cap c|=5$

I reason that it is number 1 that is correct since $\displaystyle |a \cup b \cup c| \leqslant 80 $ must be true, but this can only be said if they can belong to none as if they can't $\displaystyle |a \cup b \cup c|=80$ must be true. If they can't belong to none then I don't see this as solvable as both are wrong. I would like for someone to please confirm/contradict my thoughts on this? Have I considered every reasonable way of looking at the scenario? If it is the first, does this mean that 20 people belong to none of the groups?