QUES HON THREE (20 MAI UNDI 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

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 />Let's denote:<br />- F as the set of students studying French<br />- G as the set of students studying German<br />- R as the set of students studying Russian<br /><br />Given information:<br />- |F| = 65 (students studying French)<br />- |G| = 45 (students studying German)<br />- |R| = 42 (students studying Russian)<br />- |F ∩ G| = 20 (students studying both French and German)<br />- |F ∩ R| = 25 (students studying both French and Russian)<br />- |G ∩ R| = 15 (students studying both German and Russian)<br />- |F ∩ G ∩ R| = 8 (students studying all three languages)<br /><br />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 = |F| + |G| + |R| - 2 * (|F ∩ G| + |F ∩ R| + |G ∩ R|) + 3 * |F ∩ G ∩ R|<br /><br />Substituting the given values:<br />Number of students studying exactly one language = 65 + 45 + 42 - 2 * (20 + 25 + 15) + 3 * 8<br />Number of students studying exactly one language = 152 - 2 * 60 + 24<br />Number of students studying exactly one language = 152 - 120 + 24<br />Number of students studying exactly one language = 56<br /><br />Therefore, the number of students studying exactly one language is 56.