Answer
The Venn diagram,
Work Step by Step
(a)
In the Venn diagram, the region I represents the students registered only for a math course. So, the required number of students is
\[\begin{align}
& n\left( \text{only for math} \right)=n\left( \text{I} \right) \\
& =35
\end{align}\]
(b)
In the Venn diagram, the region III represents the students registered only for an English course. So, the required number of students is
\[\begin{align}
& n\left( \text{only for English} \right)=n\left( \text{III} \right) \\
& =25
\end{align}\]
(c)
In the Venn diagram, the sum of the regions I, II, and III represents the students registered for a math or an English course. So, the required number of students is
\[\begin{align}
& n\left( \text{math or English} \right)=n\left( \text{I} \right)+n\left( \text{II} \right)+n\left( \text{III} \right) \\
& =35+40+25 \\
& =100
\end{align}\]
(d)
In the Venn diagram, the region IV represents the number of students did not register for either a math course or an English course. So, the required number of students is
\[\begin{align}
& n\left( \text{none course} \right)=n\left( \text{IV} \right) \\
& =20
\end{align}\]
