Quick method to find $|H \cap \overline{B} \cap \overline{K}|$ starting from $|H \cap B \cap K|$? Suppose sets $S, H, K, B$. Let
$$\begin{align}
&|S| = 100 \\
&|H| = 57 \\
&|K| = 77 \\
&|B| = 66 \\
&|H\cap B| = 30 \\
&|H \cap K| = 40 \\
&|B \cap K| = 50
\end{align}$$.
The question asks to find 


*

*$|H \cap \overline{B} \cap \overline{K}|$

*$|K \cap \overline{B} \cap \overline{H}|$

*$|B \cap \overline{H} \cap \overline{K}|$


By inclusion-exclusion, 
$$\begin{align}
&|H \cup B \cup K| = |H| + |B| + |K| - (|H \cap B| + |H \cap K| + |B \cap K) + |H \cap B \cap K| \\
&|H \cap B \cap K| =|H \cup B \cup K| - (|H| + |B| + |K|) + (|H \cap B| + |H \cap K| + |B \cap K)  \\
\end{align}$$
Is there a short way to find, for example, $|H \cap \overline{B} \cap \overline{K}|$, starting from  $|H \cap B \cap K|$? 
I did the following work to find $|H \cap \overline{B} \cap \overline{K}|$, but
 I am looking for a quicker way to do this.
$$\begin{align}
&H = H \cap ((\overline{B} \cap \overline{K}) \cup (B \cup K))  \\
&H = (H \cap (\overline{B} \cap \overline{K})) \cup (H \cap (B \cup K)) \\
& (H \cap (\overline{B} \cap \overline{K})) = H -  (H \cap (B \cup K)) \\
& H \cap \overline{B} \cap \overline{K} = H - (H \cap B) \cup (H \cap K) \\
& |H \cap \overline{B} \cap \overline{K}| = |H| - (|(H \cap B)| + |(H \cap K)|) + |H \cap K \cap B| \\
& = 57 - (30 + 40) + 20 = 7 \\
\end{align}$$

I think that the quickest way is to draw a Venn diagram, calculate $|H\cap K\cap B|=20$, and fill in the cardinalities of the eight regions into which the diagram divides $S$. I very quickly get this:

I can now read off the answers; for instance, $|H\cap\overline{B}\cap\overline{K}|=7$. (In fact all three are $7$.)
