I'm trying to solve the following problem:
16 skiers that were going to compete on the next Olympic Games were asked in what disciplines will they compete. There will be two more skiers in slalom and giant slalom, than in slalom, giant slalom, and super-G. In giant slalom and super-G, there will be 3 more skiers than skiers competing in all 3 disciplines. Nobody is competing in only slalom and super-G. Only in slalom and only in super-G there are two more skiers than skiers competing in all 3 disciplines, and only in giant slalom, there will be 3 more than skiers competing in all 3 disciplines.
How many skiers compete in individual disciplines, if each of the questioned skiers in competing in at least one of the discipline.
Here's how I defined the problem:
S = {skiers competing in slalom}
V = {skiers competing in giant slalom}
G = {skiers competing in super-G}
$|S \cup V \cup G| = 16$
$|S| = 2 + |S \cap V \cap G|$
$|V| = 3 + |S \cap V \cap G|$
$|G| = 2 + |S \cap V \cap G|$
$|S \cap V| = 2 + (|S| + |V| + |G|)$
$|S \cap G| = 0 $
$|V \cap G| = 3 + |S \cap V \cap G|$
Knowing that the principle of inclusion-exclusion states: $|A \cup B \cup C| = |A| + |B| + |C| + |A \cap B|- |A \cap C| - |B \cap C| + |A \cap B \cap C|$,
I substituted the values:
16 = 2 + $|S \cap V \cap G|$ + 3 + $|S \cap V \cap G|$ + 2 + $|S \cap V \cap G|$ - (2 + 2 + $|S \cap V \cap G|$ + 3 + $|S \cap V \cap G|$ + 2 + $|S \cap V \cap G|$) - 0 - ( 3 + 2 + $|S \cap V \cap G|$ + 3 + $|S \cap V \cap G|$ + 2 + $|S \cap V \cap G|$) + 2 + $|S \cap V \cap G|$ + 3 + $|S \cap V \cap G|$ + 2 + $|S \cap V \cap G|$
For better perception, here's what it would look like if I substituted $|S \cap V \cap G|$ with x:
16 = 2+x + 3+x + 2+x -(2 + 2+x +2 + 3+x + 2 + 2+x ) - 0 - ( 3+ x ) + x
When I try to solve it, I get a false answer (no x left).
I would really appreciate if someone would point me where I went wrong.