1. Finite set problem

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

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

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

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

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

I reason that it is number 1 that is correct since $|a \cup b \cup c| \leqslant 80$ must be true, but this can only be said if they can belong to none as if they can't $|a \cup b \cup c|=80$ 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?

2. Your reasoning follows using inclusion-exclusion. I would say that the wording of the problem statement ("each can belong to any number of a, b, c") implies that that a participant can belong to none of them, since zero is a number.

3. |aUbUc| = |a| + |b| + |c| - |a∩b| - |a∩c| - |b∩c| + |a∩b∩c|.

(1) implies 80 ≥ 50 + 40 + 30 - 30 - 20 - 20 + 10 = 60, which is true.

(2) implies 80 ≥ 50 + 40 + 30 - 10 - 10 - 20 + 5 = 85, which is false.

yes, 20 people belong to none of the 3 sets.

4. Originally Posted by Deveno
|aUbUc| = |a| + |b| + |c| - |a∩b| - |a∩c| - |b∩c| + |a∩b∩c|.
Yes, this is what I was using. Thanks.