Question

In a survey of 60 people, 30 speak Bengali, 20 speak Malayalam, and 15 speak Tamil. 10 people do not speak any of the three languages, and 5 speak all three. Find out the number of people who speak exactly two of the three languages.

Answer 1

$\large \tt45 \: \: people \: \: \: speak \: \: 2 \: \: languages$

Answer 2

Concept

Probability is really how possibly some thing is to happen.The probability of any event depends on the number of favorable results and the total of the results.

Given

There are total $60$ people, people who speak Bengali is $30,$ people who speak Malayalam is $20$, and people who speak Tamil is $15$ and there are $10$ people who do not speak any language and people who speak three languages ​​are  $5$.

Find

We have to find the number of people who exactly speak any two languages out of three.

Solution

Let $B$ be the set of people who speak Bengali, $M$ be the set of people who speak Malayalam and $T$ be the people who speak Tamil.

The number of people who speaks three languages

$60-n(B\cup \cup T)=10$

$\Rightarrow n(B\cup M\cup T)=50$

Therefore, $n(B)=30,n(M)=20,n(T)=15$, $n(B\cap M\cap T)=5$

and $n(B\cup M\cup T)=10$.

Now, the number of people who speak exactly two of the three languages is $n(B\cap M)+n(M\cap T)+n(T\cap B)-3n(B\cap M\cap T)$

$n(B\cup M\cup T)=n(B)+n(M)+n(T)-[n(B\cap M)+n(M\cap T)+n(T\cap B)]+n(B\cap M\cap T)$

$\Rightarrow n(B\cap M)+n(M\cap T)+n(T\cap B)=n(B)+n(M)+n(T)+n(B\cap M\cap T)-n(B\cup M\cup T)$

Subtract $3n(B\cap M\cap T)$ from both sides, we get

$n(B\cap M)+n(M\cap T)+n(T\cap B)-n(B\cap M\cap T)=n(B)+n(M)+n(T)+n(B\cap M\cap T)-n(B \cup M\cup T)-3n(B\cap M\cap T)$

$\Rightarrow n(B\cap M)+n(M\cap T)+n(T\cap B)-3n(B \cap M \cap T) =30+20+15+5-50-3\times 5$

$\Rightarrow n(B\cap M)+n(M\cap T)+n(T\cap B)-3n(B \cap M \cap T)=5$

Hence, the number of people who speak exactly two languages out of three is $5$.

#SPJ2