SOLUTION: What does the phrase "...at least two people are taking two languages at once" mean, in as set problems?

Question 621244: What does the phrase "...at least two people are taking two languages at once" mean, in as set problems?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
First, it means you have to pay close attention to the wording.
Sometimes we assume that something was meant, but it was not specifically stated.

More specifically, "...at least two people are taking two languages at once" means some intersections of two sets include some students, with a total of 2 or more in those intersections.
There could be many different situations. Maybe one student belongs to the set of people taking French class and to the set of people taking German, another student is in French and Italian, and there may be others equally ambitious that are also in the same or other intersections of 2 or more sets. You may not have enough information to state a result with an "equal to".
Without more information, you may end up with an inequality for your answer.

If they tell you that a certain number of students is taking French and a certain number is taking Spanish and the school does not offer courses in any other language, and you just add the numbers given, you would have 2 or more students counted twice. The number of students taking at least one language would be the sum of the numbers given, minus the number "taking two languages at once."

If m students take language M, n students take language N, x students take language X, y students take Y and z students take Z, and d students are "taking two languages at once" the number of students taking at least one language would be m+n+x+y+z-d. And if "at least two people are taking two languages at once" so the number of students taking at least one language would be
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