Mathematics, Science and English

BEGINNER'S METHOD

Let

$\begin{aligned} M & =\{ Pupils\quad who\quad study\quad Mathematics\} \\ S & =\{ Pupils\quad who\quad study\quad Science\} \\ E & =\{ Pupils\quad who\quad study\quad English\} \end{aligned}$

Since every pupil must study at least one of the three subjects, we have $n(M\cup S\cup G)=n(\varepsilon )=40$ .

Let the number of pupils who study all three subjects be $x$ . Hence, the central region in the above figure is marked $x$ . Next, the different values of $a$ , $b$ , $c$ , $d$ , $e$ and $f$ would be found to complete the Venn diagram.

Clarification:

$\begin{aligned} x & =M\cap S\cap E \\ a & =M\cap S\cap E^{ C } \\ b & =M\cap E\cap S^{ C } \\ c & =S\cap E\cap M^{ C } \\ d & =M\cap (S\cap E)^{ C } \\ e & =E\cap (S\cap M)^{ C } \\ f & =S\cap (E\cap M)^{ C } \\ & \end{aligned}\\$

Since $12$ pupils are studying Mathematics and Science, there must be $12-x$ pupils who are studying only these two subjects.

By the same argument,

$b=15-x$ and $c=14-x$

Altogether, $20$ pupils study Mathematics.

$\begin{aligned} \therefore \quad d&=20-x-a-b\\ & =20-x-(12-x)-(15-x)\\ & =x-7 \end{aligned}$

By the same argument,

$\begin{aligned} e&=28-x-(15-x)-(14-x)\\ & =x-1\\ f&=22-x-(12-x)-(14-x)\\ & =x-7 \end{aligned}$

We know,

$\begin{aligned} n(M\cup S\cup E)=d+b+a+x+f+c++e&=40 \\ \Longrightarrow \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 29+x &=40 \\ \therefore \quad x &=\boxed{11} \end{aligned}$