Question

Suppose that $$100$$ of $$120$$ mathematics students at a College take at least one of the languages French, German and Russian. Also suppose $$65$$ study French, $$45$$ study German, $$42$$ study Russian, $$20$$ study French and German, $$25$$ study French and Russian, $$15$$ studies German and Russian. Find the number of students studying all subjects. Also, find the bar studying exactly one subject.

Answer 1

$$ {\textbf{Step - 1: Draw Venn Diagram and label the given information}} $$

$$ {\text{The provided information are, }} $$

$$ {\text{65 students study French}} $$

$$ {\text{45 study German}} $$

$$ 42{\text{ study Russian}} $$

$$ {\text{20 study both French(f) and German(g)}} $$

$$ {\text{25 study both French(f) and Russian(r)}} $$

$$ {\text{15 study both German(g) and Russian(r)}} $$

$$ {\text{n(French) = 65}} $$

$$ {\text{n(French }} \cap {\text{ German) = 20}} $$

$$ {\text{n(French }} \cap {\text{ Russian) = 25}} $$

$$ {\text{n(German }} \cap {\text{ Russian) = 15}} $$

$$ {\text{we have to find that number of student studying all subject}}$$

$${\text{ and number of student studying only one subject}}{\text{.}} $$

$$ {\text{number of student studying all subject = n(f}} \cup {\text{g}} \cup {\text{r)}} $$

$$ {\text{number of student studying only one subject = n(f}} \cap {\text{g}} \cap {\text{r)}} $$

$$ {\textbf{Step - 2: Calculation}} $$

$$ {\text{n(f}} \cup {\text{g}} \cup {\text{r)}} = {\text{n(f)}} + {\text{n(g)}} + {\text{n(r)}} - {\text{n(f}} \cap {\text{g)}} - {\text{n(f}} \cap {\text{r)}} - {\text{n(g}} \cap {\text{r)}} + {\text{n(f}} \cap {\text{g}} \cap {\text{r)}} $$

$$ \Rightarrow {\text{100 = 65 + 45 + 42 - 20 - 25 - 15 + n(f}} \cap {\text{g}} \cap {\text{r)}} $$

$$ \Rightarrow {\text{n(f}} \cap {\text{g}} \cap {\text{r)}} = 100 - 92 $$

$$ \Rightarrow {\text{n(f}} \cap {\text{g}} \cap {\text{r)}} = 8 $$

$$ {\text{So, 8 student study all subject}} $$

$$ {\text{and number of student study only one subject = 28 + 18 + 10 = 56}} $$

$$ {\textbf{Hence, number of students study all subject is 8 and students study exactly one subject}{\textbf{ }}\\{\textbf{ is 56}}}{\text{.}} $$