Draw three overlapping circles and label them "F" (for "French"), "G" (for "German"), and "S" (for "Spanish").
You are told "There are 29 students who take at least 2 languages at once." So the total number of students who would fit into the overlaps of those circles is 29. You are not told how many take, say, "French and German but not Spanish" or "all three languages" so enter "a" where "S" and only "F" overlap, "b" where only "G" and "F" overlap, "c" where only "S" and "G" overlap, and "d" where all three circles overlap. We must have a+ b+ c+ d= 29.
You are told that "there are 40 Spanish students" but that includes the "a" students who take Spanish and French, the "c" students who take Spanish and German, and the "d" students who take all three language. There are 40- a- c- d students who take Spanish only.
Similarly, you are told that there are "42 French students" so there are 42- a- b- d students who take French only.
And you are told that there are "46 German students" so there are 46- b- c- d students who take German only.
So in the 7 areas where those three circles overlap, we have "40- a- c- d", "42- a- b- d", "45- b- c- d", "a", "b", "c", and "d" where, now, each student is counted only once. Add those together and set it equal to 103 since we are told that is the number of foreign language students.
You are asked, "how many students take all three languages at once?". That is what we called "d". Are you able to find "d"?
