Answer
The number of elements for the different asked regions is shown below:
Work Step by Step
Simplify region II to get the elements:
\[n\left( A\cap B \right)\]is equal to region II\[+\]V.
\[\begin{align}
& n\left( A\cap B \right)=3 \\
& \text{II}+\text{V}=3 \\
& \text{II}+2=3 \\
& \text{II}=1
\end{align}\]
For region IV:
\[n\left( A\cap C \right)\]is equal to region IV\[+\]V.
\[\begin{align}
& n\left( A\cap C \right)=5 \\
& \text{IV}+\text{V}=5 \\
& \text{IV}+2=5 \\
& \text{IV}=3
\end{align}\]
For region VI:
\[n\left( B\cap C \right)\]is equal to region VI\[+\]V.
\[\begin{align}
& n\left( B\cap C \right)=3 \\
& \text{VI}+\text{V}=3 \\
& \text{VI}+2=3 \\
& \text{VI}=1
\end{align}\]
For region I:
\[n\left( A \right)\]is equal to region I\[+\]II\[+\]IV\[+\]V.
\[\begin{align}
& n\left( A \right)=\text{I}+\text{II}+\text{IV}+\text{V} \\
& 11=1+2+3+\text{I} \\
& \text{I}=11-6 \\
& \text{I}=5
\end{align}\]
For region III:
\[n\left( B \right)=8\]is equal to region II\[+\]III\[+\]V\[+\]VI.
So,
\[\begin{align}
& \text{II}+\text{III}+\text{V}+\text{VI}=n\left( B \right) \\
& 1+2+1+\text{III}=8 \\
& \text{III}=8-4 \\
& \text{III}=4
\end{align}\]
For region VII:
\[n\left( C \right)=14\]is equal to region IV\[+\]V\[+\]VI\[+\]VII.
\[\begin{align}
& \text{IV}+\text{V}+\text{VI}+\text{VII}=n\left( C \right) \\
& 3+2+1+\text{VII}=14 \\
& \text{VII}=14-6 \\
& \text{VII}=8
\end{align}\]
For region VIII:
\[\begin{align}
& \text{VIII}=30-n\left( A\cup B\cup C \right) \\
& =30-\left( 5+1+4+3+2+1+8 \right) \\
& =30-24 \\
& =6
\end{align}\]
