# In a survey of 100 students, the number of students studying the various languages were found to be :

Question:

In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find:

(i) How many students were studying Hindi?

(ii) How many students were studying English and Hindi?

Solution:

Let E, H and S be the sets of students who study English, Hindi and Sanskrit, respectively.

Also, let U be the universal set.

Now, we have:

$n(\mathrm{E})=26, n(\mathrm{~S})=48, n(\mathrm{E} \cap \mathrm{S})=8$ and $n(\mathrm{~S} \cap \mathrm{H})=8$

Also,

$n\left(\mathrm{E} \cap \mathrm{H}^{\prime}\right)=23$

$\Rightarrow n(\mathrm{E})-n(\mathrm{E} \cap \mathrm{H})=23$

$\Rightarrow 26-n(\mathrm{E} \cap \mathrm{H})=23$

$\Rightarrow n(\mathrm{E} \cap \mathrm{H})=3$

Therefore, the number of students studying English and Hindi is 3

$n\left(\mathrm{E} \cap \mathrm{H}^{\prime} \cap \mathrm{S}^{\prime}\right)=18$

$\Rightarrow n(\mathrm{E})-n\left\{\mathrm{E} \cap(\mathrm{H} \cup \mathrm{S})^{\prime}\right\}=18$

$\Rightarrow 26-n\{(\mathrm{E} \cap \mathrm{H}) \cup(\mathrm{E} \cap \mathrm{S})\}=18$

$\Rightarrow 26-\{3+8-n(\mathrm{E} \cap \mathrm{H} \cap \mathrm{S})\}=18$

$\Rightarrow n(\mathrm{E} \cap \mathrm{H} \cap \mathrm{S})=3$

Also,

$n\left(\mathrm{E}^{\prime} \cap \mathrm{H}^{\prime} \cap S^{\prime}\right)=24$

$\Rightarrow n(\cup)-n(\mathrm{E} \cup \mathrm{H} \cup S)=24$

$\Rightarrow n(\mathrm{E} \cup \mathrm{H} \cup S)=76$

$\therefore$ Number of students studying Hindi $=n(\mathrm{E} \cup \mathrm{H} \cup \mathrm{S})-n(\mathrm{E})-n(\mathrm{~S})+n(\mathrm{E} \cap \mathrm{H})+n(\mathrm{E} \cap \mathrm{S})+n(\mathrm{~S} \cap \mathrm{H})-n(\mathrm{E} \cap \mathrm{H} \cap \mathrm{S})$

$=76-24-48+3+8+8-3$

$=18$