Answer
The Venn diagram,
Work Step by Step
(a)
In the Venn diagram, the region V represents the students who listened to only rock music. So, the required number of students is
\[\begin{align}
& n\left( \text{only rock} \right)=n\left( \text{V} \right) \\
& =23
\end{align}\]
(b)
In the Venn diagram, the region III represents the students who listened to classical and jazz, but not rock. So, the required number of students is
\[\begin{align}
& n\left( \text{only classical and jazz} \right)=n\left( \text{IV} \right) \\
& =3
\end{align}\]
(c)
In the Venn diagram, the region II represents the students who listened to classical or jazz, but not rock. So, the required number of students is
\[\begin{align}
& n\left( \text{classical or jazz, but not rock} \right)=n\left( \text{I} \right)+n\left( \text{IV} \right)+n\left( \text{VII} \right) \\
& =12+3+17 \\
& =32
\end{align}\]
(d)
In the Venn diagram, the sum of the regions I, V, and VII represents the number of students who listened to exactly one type of the musical styles. So, the required number of students is
\[\begin{align}
& n\left( \text{exactly one style} \right)=n\left( \text{I} \right)+n\left( \text{V} \right)+n\left( \text{VII} \right) \\
& =12+23+17 \\
& =52
\end{align}\]
(e)
In the Venn diagram, the sum of the regions II, III, IV, and VI represents the number of students listened to at least two types of the musical styles. So, the required number of students is
\[\begin{align}
& n\left( \text{at least two styles} \right)=n\left( \text{II} \right)+n\left( \text{III} \right)+n\left( \text{IV} \right)+n\left( \text{VI} \right) \\
& =5+7+3+7 \\
& =22
\end{align}\]
(f)
In the Venn diagram, the region VIII represents the number of students who did not listen to any of the musical styles. So, the required number of students is
\[\begin{align}
& n\left( \text{none} \right)=n\left( \text{VIII} \right) \\
& =6
\end{align}\]
