Draw a picture: Draw 3 circles, overlapping. Each circle represents the students that are taking that particular course. Label each region as "Math", "German", and "English. In that region where all three circles overlap write the number 15 to represent the 15 students that are taking all three courses. In the region where the "English" and "Math" circles overlap write "x" representing the (unknown) students taking those two courses. In the region where "Math" and "German" overlap, write "y" representing the (unknown) students taking those two courses. In the region where the "English" and "German" circles overlap, write "z" representing the (unknown) students taking those two courses. Since there are a total of 40 students taking math there must be 40- (x+ y+ 15)= 25- x- y in the final open region of the "Math" circle- representing those student who are taking only math. Since there are 35 students taking English, there must be 35- (x+z+ 15)= 20- x- z in the final open region of the "English" circle- representing those students who are taking only English. Since there are 30 students taking German, the must be 30- (y+ z+ 15)= 15- y- z in the final open region of the "German" circle- representing those students who are taking only German.

The point of all that is that each student is counted in thatonly once. We can get the total number of students by adding those numbers:

(25-x-y)+ (20-x-z)+ (15-y-z)+ (x+ y+ z)+ 15= 75- (x+ y+ z) and we know there are a total of 70 students!

where , "y", and "z" representing the (unknown) students taking exactly two of the courses. In the final 3 regions write