Answer
The number of elements for the different asked regions is shown below:
Work Step by Step
Simplify region II:
\[n\left( A\cap B \right)=17\]is equal to region II\[+\]V.
\[\begin{align}
& \text{II}+\text{V}=17 \\
& \text{II}+5=17 \\
& \text{II}=12
\end{align}\]
Then, for region IV:
\[n\left( A\cap C \right)=11\]is equal to region IV\[+\]V.
\[\begin{align}
& \text{IV}+\text{V}=11 \\
& \text{IV}+5=11 \\
& \text{IV}=6
\end{align}\]
Then, for region VI:
\[n\left( B\cap C \right)=9\]is equal to region VI\[+\]V.
\[\begin{align}
& \text{VI}+\text{V}=9 \\
& \text{VI}+5=9 \\
& \text{VI}=4
\end{align}\]
Then, for region I:
\[n\left( A \right)=26\]is equal to region I\[+\]II\[+\]IV\[+\]V.
\[\begin{align}
& \text{I}+\text{II}+\text{IV}+\text{V}=26 \\
& 12+6+5+\text{I=26} \\
& \text{I=26}-23 \\
& \text{I}=3
\end{align}\]
Then, for region III:
\[n\left( B \right)=22\]is equal to region II\[+\]III\[+\]V\[+\]VI.
\[\begin{align}
& \text{II}+\text{III}+\text{V}+\text{VI}=22 \\
& 12+5+4+\text{III=22} \\
& \text{III=22}-21 \\
& \text{III}=1
\end{align}\]
Then, for region VII:
\[n\left( C \right)=25\]is equal to region IV\[+\]V\[+\]VI\[+\]VII.
\[\begin{align}
& \text{IV}+\text{V}+\text{VI}+\text{VII}=25 \\
& 6+5+4+\text{VII=25} \\
& \text{VII=25}-15 \\
& \text{VII}=10
\end{align}\]
Then, for region VIII:
\[\begin{align}
& \text{VIII}=42-n\left( A\cup B\cup C \right) \\
& =42-\left( 3+12+1+6+5+4+10 \right) \\
& =42-41 \\
& =1
\end{align}\]
