Out of a group of 120 students, the following information is given 65 students study French 45 students study German 42 students study Russian 20 students study French and German 25 students study French and Russian 15 students study German and Russian 8 students study all the 3 languages a) Draw a venn diagram for the above information b) Find the number of students studying exactly one language c) Determine the number of students who take none of the languages d) Find the number of students studying exactly two languages

a) To draw a Venn diagram for the given information, we can represent each language as a circle in the diagram. The intersection of the circles represents the students studying multiple languages.<br /><br />b) To find the number of students studying exactly one language, we need to subtract the number of students studying multiple languages from the total number of students studying each language. <br /><br />Number of students studying exactly one language = Total number of students studying each language - Number of students studying multiple languages<br /><br />For French: 65 - 20 - 25 - 8 = 12<br />For German: 45 - 20 - 15 - 8 = 2<br />For Russian: 42 - 25 - 15 - 8 = 4<br /><br />So, the number of students studying exactly one language is 12 (French) + 2 (German) + 4 (Russian) = 18.<br /><br />c) To determine the number of students who take none of the languages, we need to subtract the number of students studying at least one language from the total number of students.<br /><br />Number of students studying at least one language = Total number of students - Number of students studying none of the languages<br /><br />Number of students studying none of the languages = Total number of students - Number of students studying at least one language<br /><br />Number of students studying none of the languages = 120 - (65 + 45 + 42 - 20 - 25 - 15 - 8) = 120 - 90 = 30<br /><br />So, the number of students who take none of the languages is 30.<br /><br />d) To find the number of students studying exactly two languages, we need to subtract the number of students studying all three languages from the number of students studying each pair of languages.<br /><br />Number of students studying exactly two languages = Number of students studying each pair of languages - Number of students studying all three languages<br /><br />For French and German: 20 - 8 = 12<br />For French and Russian: 25 - 8 = 17<br />For German and Russian: 15 - 8 = 7<br /><br />So, the number of students studying exactly two languages is 12 (French and German) + 17 (French and Russian) + 7 (German and Russian) = 36.