Question

Out of a group of 200 students (Who know at least one language), $100$ students know English, \[80\] students know Kannada, $70$ students know Hindi. If $20$ students know all the three languages. Find the number of students who know exactly two languages.

Answer 1

Hint: To solve this question, we will use the formula of set theory that is $n\left( A\bigcup B\bigcup C \right)=n\left( A \right)+n\left( B \right)+n\left( C \right)-\left[ n\left( A\bigcap B \right)+n\left( B\bigcap C \right)+n\left( C\bigcap A \right) \right]+n\left( A\bigcap B\bigcap C \right)$, where $n\left( A \right),n\left( B \right),n\left( C \right)$ are number of students who know one language only, $n\left( A\bigcap B \right),n\left( B\bigcap C \right),n\left( C\bigcap A \right)$ are number of students who know two languages, $n\left( A\bigcap B\bigcap C \right)$ is number of students who know three languages and $n\left( A\bigcup B\bigcup C \right)$ is number of all students who know any languages. We will substitute corresponding values and will simplify it to get the required answer.

Complete step by step solution:
Let consider that $A$ is set of students who know English, $B$ be the set of the students who know Kannada and $C$ is set of students who know Hindi. So, from the questions, we have:
$\begin{align}
  & \Rightarrow n\left( A \right)=100 \\
 & \Rightarrow n\left( B \right)=80 \\
 & \Rightarrow n\left( C \right)=70 \\
\end{align}$
Total number of students, $n\left( A\bigcup B\bigcup C \right)=200$
And number of students who know all the three languages, $n\left( A\bigcap B\bigcap C \right)=20$
Now, we will use the related formula.
$\Rightarrow n\left( A\bigcup B\bigcup C \right)=n\left( A \right)+n\left( B \right)+n\left( C \right)-\left[ n\left( A\bigcap B \right)+n\left( B\bigcap C \right)+n\left( C\bigcap A \right) \right]+n\left( A\bigcap B\bigcap C \right)$
Here, we will substitute the corresponding values as:
$\Rightarrow 200=100+80+70-\left[ n\left( A\bigcap B \right)+n\left( B\bigcap C \right)+n\left( C\bigcap A \right) \right]+20$
We will get $250$ after adding $100,80$ and $70$ as:
$\Rightarrow 200=250-\left[ n\left( A\bigcap B \right)+n\left( B\bigcap C \right)+n\left( C\bigcap A \right) \right]+20$
Now, we will get $270$ when we will add $250$ and $20$ as:
$\Rightarrow 200=270-\left[ n\left( A\bigcap B \right)+n\left( B\bigcap C \right)+n\left( C\bigcap A \right) \right]$
After changing the places in the above step, we can write the above step as:
$\Rightarrow \left[ n\left( A\bigcap B \right)+n\left( B\bigcap C \right)+n\left( C\bigcap A \right) \right]=270-200$
Here, we will do the subtraction and will get $70$ after subtracting $200$ from $270$ as:
$\Rightarrow \left[ n\left( A\bigcap B \right)+n\left( B\bigcap C \right)+n\left( C\bigcap A \right) \right]=70$
So, the total number of students who know two languages is $70$.

Note: Here, we will check whether the solution is correct or not by using the formula.
$\Rightarrow n\left( A\bigcup B\bigcup C \right)=n\left( A \right)+n\left( B \right)+n\left( C \right)-\left[ n\left( A\bigcap B \right)+n\left( B\bigcap C \right)+n\left( C\bigcap A \right) \right]+n\left( A\bigcap B\bigcap C \right)$
Substitute the corresponding values in the formula as:
$\Rightarrow 200=100+80+70-70+20$
Here, we will cancel out the equal like term and will do the addition as:
$\begin{align}
  & \Rightarrow 200=180+20 \\
 & \Rightarrow 200=200 \\
\end{align}$
Since, $L.H.S.=R.H.S.$
Hence, the solution is correct.