Question

A class has ,$175$ students. the following table shows the number of students studying one or more of the following subjects in this class. find how many students are enrolled in mathematics alone, physics alone, and chemistry alone? are there students who have not been offered any of these three subjects?

Subject	Number of students
Mathematics	100
Physics	70
Chemistry	46
Mathematics and Physics	30
Mathematics and Chemistry	28
Physics and Chemistry	23
Mathematics, Physics and Chemistry	18

Answer 1

Step 1.Find the number of students who studied maths or physics or chemistry

From the given table,

Total number of students, $n\left(U\right)=175$

Total number of students enrolled in Mathematics alone, $n\left(M\right)=100$

Total number of students enrolled in Physics alone, $n\left(P\right)=70$

Total number of students enrolled in Chemistry alone, $n\left(C\right)=46$

So that, $n\left(U\right)=175,n\left(M\right)=100,n\left(P\right)=70\mathrm{and}n\left(C\right)=46$

$n(M\cap P)=30,n(M\cap C)=28,n(P\cap C)=23,andn(M\cap P\cap C)=18$

The number of students who studied maths or physics or chemistry is given by $n(M\cup P\cup C)$ .

$$\begin{gathered} \therefore n(M\cup P\cup C)=n\left(M\right)+n\left(P\right)+n\left(C\right)-n(M\cap P)-n(M\cap C)-n(P\cap C)+n(M\cap P\cap C) \\ \Rightarrow =100+70+46-30-28-23+18 \\ \Rightarrow =234-81 \\ \Rightarrow =153 \end{gathered}$$

Step 2. Calculate the total number of students enrolled in Mathematics alone.

The number of students enrolled in Mathematics alone is computed as,

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{students}=n\left(M\right)-n(M\cap P)-n(M\cap C)+n(M\cap P\cap C) \\ \Rightarrow =100-30-28+18 \\ \Rightarrow =118-58 \\ \Rightarrow =60 \end{gathered}$$

Thus, the total $60$ students enrolled in Mathematics alone.

Step 3. Calculate the total number of students enrolled in Physics alone.

The number of students enrolled in physics alone is computed as,

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{students}=n\left(P\right)-n(M\cap P)-n(P\cap C)+n(M\cap P\cap C) \\ \Rightarrow =70-30-23+18 \\ \Rightarrow =88-53 \\ \Rightarrow =35 \end{gathered}$$

Thus, the total $35$ students enrolled in physics alone.

Step 4. Calculate the total number of students enrolled in Chemistry alone.

The number of students enrolled in Chemistry alone is computed as,

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{students}=n\left(C\right)-n(M\cap C)-n(P\cap C)+n(M\cap P\cap C) \\ \Rightarrow =46-28-23+18 \\ \Rightarrow =64-51 \\ \Rightarrow =13 \end{gathered}$$

Thus, the total $13$ students enrolled in chemistry alone.

Step 5. Calculate the number of students who have not been offered any of these three subjects.

The number of students who have not been offered any of these three subjects is computed as,

$\mathrm{Number}\mathrm{of}\mathrm{students}=175-153=22$

Thus, there are $22$ students who have not been offered any of these three subjects.

Hence,

The total $60$ students enrolled in Mathematics alone.

The total $35$ students enrolled in physics alone.

The total $13$ students enrolled in chemistry alone.

There are $22$ students who have not been offered any of these three subjects.