Answer
The number of elements for the different asked regions is shown below:
Work Step by Step
Simplify region II:
\[n\left( A\cap B \right)=6\]is equal to region II\[+\]V.
\[\begin{align}
& \text{II}+\text{V}=6 \\
& \text{II}+2=6 \\
& \text{II}=4
\end{align}\]
For region IV:
\[n\left( A\cap C \right)=7\]is equal to region IV\[+\]V.
\[\begin{align}
& \text{IV}+\text{V}=7 \\
& \text{IV}+2=7 \\
& \text{IV}=5
\end{align}\]
For region VI:
\[n\left( B\cap C \right)=8\]is equal to region VI\[+\]V.
\[\begin{align}
& \text{VI}+\text{V}=8 \\
& \text{VI}+2=8 \\
& \text{VI}=6
\end{align}\]
For region I:
\[n\left( A \right)=21\]is equal to region I\[+\]II\[+\]IV\[+\]V.
\[\begin{align}
& \text{I}+\text{II}+\text{IV}+\text{V}=21 \\
& 4+2+5+\text{I}=21 \\
& \text{I}=21-11 \\
& \text{I}=10
\end{align}\]
For region III:
\[n\left( B \right)=15\]is equal to region II\[+\]III\[+\]V\[+\]VI.
\[\begin{align}
& \text{II}+\text{III}+\text{V}+\text{VI}=15 \\
& 4+2+6+\text{III}=15 \\
& \text{III}=15-12 \\
& \text{III}=3
\end{align}\]
For region VII:
\[n\left( C \right)=14\]is equal to region IV\[+\]V\[+\]VI\[+\]VII.
\[\begin{align}
& \text{IV}+\text{V}+\text{VI}+\text{VII}=14 \\
& 5+2+6+\text{VII}=14 \\
& \text{VII}=14-13 \\
& \text{VII}=1
\end{align}\]
For region VIII:
\[\begin{align}
& \text{VIII}=32-n\left( A\cup B\cup C \right) \\
& =32-\left( 10+5+4+3+2+6+1 \right) \\
& =32-31 \\
& =1
\end{align}\]
