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.
\[\begin{align}
& \text{II}+\text{V}=17 \\
& \text{II}+7=17 \\
& \text{II}=10
\end{align}\]
For region IV:
\[n\left( A\cap C \right)=11\]is equal to region IV\[+\]V.
\[\begin{align}
& \text{IV}+\text{V}=11 \\
& \text{IV}+7=11 \\
& \text{IV}=4
\end{align}\]
For region VI:
\[n\left( B\cap C \right)=8\]is equal to region VI\[+\]V.
\[\begin{align}
& \text{VI}+\text{V}=8 \\
& \text{VI}+7=8 \\
& \text{VI}=1
\end{align}\]
For region I:
\[n\left( A \right)=26\]is equal to region I\[+\]II\[+\]IV\[+\]V.
\[\begin{align}
& \text{I}+\text{II}+\text{IV}+\text{V}=n\left( A \right) \\
& 10+4+7+\text{I}=26 \\
& \text{I}=26-21 \\
& \text{I}=5
\end{align}\]
For region III:
\[n\left( B \right)=21\]is equal to region II\[+\]III\[+\]V\[+\]VI.
\[\begin{align}
& \text{II}+\text{III}+\text{V}+\text{VI}=n\left( B \right) \\
& 10+7+1+\text{III}=21 \\
& \text{III}=21-18 \\
& \text{III}=3
\end{align}\]
For region VII:
\[n\left( C \right)=18\]is equal to region IV\[+\]V\[+\]VI\[+\]VII.
\[\begin{align}
& \text{IV}+\text{V}+\text{VI}+\text{VII}=n\left( C \right) \\
& 4+7+1+\text{VII}=18 \\
& \text{VII}=18-12 \\
& \text{VII}=6
\end{align}\]
For region VIII:
\[\begin{align}
& \text{VIII}=38-n\left( A\cup B\cup C \right) \\
& =38-\left( 5+10+4+3+7+1+6 \right) \\
& =38-36 \\
& =2
\end{align}\]
