Q:

How do I interpret b and c in this union of sets question?

I've been set the question:
(2) Let A,B, and C be sets with the following properties:(a)|A|= 100,|B|= 50,and|C|= 48.(b) The number of elements that belong to exactly one of the three sets is twice the number that belong to exactly two of the sets.(c) The number of elements that belong to exactly one of the three sets is three times the number that belong to all of the sets.How many elements belong to all three sets?
I'm having a hard time interpreting b and c, as b could imply whats exclusively in A (or B or C) = 2|BnC| (or with others) or |A|= 2|BnC|, and C doesn't use exactly in the final part i.e. 'exactly' all three sets what would imply |AnBnC| however it doesn't do this hence I'm left wondering what the use of exactly would mean in the rest of the question, with not much I can do apart from working out all possible ways to do this and assume one is correct in the end. Any help would be appreciated.

A:

What I like to do is divide it into 8 distinct disjoint all encompassing sets.  They are:
1) $A\cap B \cap C$.    These are the only elements in all three sets. This is one of the variables we are asked about.  Lets say there are $x$ elements. This are also some of the elments that compose $A$, $B$ or $C$.

$x =|A\cap B\cap C|$ is the element of elements in all three set. .

2) $A\cap B \cap C^c$. ($C^c$ means $C$'s complement.  All the elements not in see).  These are some but not all elements that are in exactly $2$ sets.  We'll call this $d$ elements. These are also some of the elements in $A$ and $B$.
3) $A \cap B^c \cap C$.  Call this number $e$.
4) $A^c \cap B \cap C$. Call this number $f$.

Sets 2-4 are all the elements in exactly two sets.  $d+e+f=$ number of elements in exactly two sets..

5) $A^c\cap B^c \cap C$.  Call this number $g$.
6) $A^c \cap B \cap C^c$. Call this number $h$
7) $A \cap B^c \cap C^c$. Call this number $i$

Sets 5-7 are the sets with elements in exactly one set.  $g+h+i\le 100$ is the number of elements in exactly one set.

8) $A^c\cap B^c \cap C^c$.  These are the elements not in any of the sets.  This will not have anything to do with our work.

Sets 1) 2) 3) and 7) make up the elements of $A$ so $|A| = 100=x + d+e + i$.
Sets 1) 2) 4) and 6) make up the elements of $B$ so $|B| = 50  = x + d+ f + h$.
Sets 1) 3) 4) and 5) make up the elements of $C$ so $|C| = 48 = x + e+f +g$.

and we know

$g+h+i = 2(d+e +f)$
$g+h+i = 3x$.

5 equations and 7 unknowns. But we are only asked to solve for $x$, and we have restrictions.
Adding $|A| + |B|+|C|$ together we get.
$198 = 3x + 2(d+e+f) + (g+h+i)$
Substituting $2(d+e+f)=(g+h+i) = 3x$ we get
$198 = 3x + 3x  +3x = 9x$
and $x = 22$.
We aren't asked to solve anything more.

