a. The number of students studying at least one of the three languages is 100.
b. The Venn Diagram for the data is attached.
c. The number of students studying exactly one language is 56.
d. The number of students studying exactly two languages is 36.
a. The number of students studying at least one of the three languages:
Thus, the calculation would be: 65 French = 65 + 45 (German) + 42 (Russian) - 20 (French and German) - 25 (French and Russian) - 15 (German and Russian) + 8 (all three languages) = 100 students.
b. A Venn diagram has been drawn and attached here. The Venn diagram will have three overlapping circles, one for each language. The overlaps between the circles represent the students studying two languages, and the area where all three circles overlap represents the students studying all three languages.
c. The number of students studying exactly one language:
French only: 65 (total French) - 20 (French and German) - 25 (French and Russian) + 8 (all three) = 28
German only: 45 (total German) - 20 (French and German) - 15 (German and Russian) + 8 (all three) = 18
Russian only: 42 (total Russian) - 25 (French and Russian) - 15 (German and Russian) + 8 (all three) = 10
Thus, 28 + 18 + 10 = 56 students are studying exactly one language.
d. The number of students studying exactly two languages can be found by adding the number of students studying each pair of languages, and then subtracting the students studying all three languages (since they are being counted in each pair):
French and German only: 20 - 8 = 12
French and Russian only: 25 - 8 = 17
German and Russian only: 15 - 8 = 7
Thus, 12 + 17 + 7 = 36 students are studying exactly two languages.