how many students speak both 3 languages?

Question

In a class with 50 students, 30 do English, 15 do French, 19 do German, 6 do both English and German, 7 do both French and English , 5 do both French and German. How many students do all 3 languages English French and German?

Answer 1

30+15+19 = 64 is the total number of students in the 3 language classes. There are 50 total students so there must be 14 overlaps. 6 english/german students covers 6 overlaps. Similarly, 7 and 5 cover 12 overlaps. 6+12=18 given overlaps, showing 4 excess. So there must be 4 students taking all 3 subjects.

 

To double check, we state how many students are doing exactly 2 subjects. 4 students doing all three implies 2 are doing E/G, 3 doing F/E, and 1 doing F/G. The double subject students gives us 2+3+1 = 6 overlaps. The triple subject students gives us 2 overlaps each so that adds up to 6+4*2=14.