# Venn diagram involving three subsets

There are 1000 students in a class. 200 passed algebra, 100 passed statistics, and 150 passed mathematics. Only 40 passed the three courses, while equal number of students passed exactly two courses. If 6 students were absent for the examination, what are the number of students that passed exactly two courses?. And the number that failed each of the courses?

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It helps us to help you if you can show us any work you've done on the problem (e.g., organizing the information you're given, your hunches about what to do with that information, etc.), and also ask specific questions, addressing where you're stuck, etc. With respect to homework, a lot of us prefer to provide direction when needed, a few hints, etc., because just writing out an answer isn't going to help you much, in the long run. – amWhy Jun 23 '11 at 0:16
@Elizabeth: Not a great class. How many people failed everything? – André Nicolas Jun 23 '11 at 0:35
It's interesting that neither algebra nor statistics are considered as part of mathematics in this word problem. – t.b. Jun 23 '11 at 0:49
@amWhy: Actually, I was asking because with current wording the problem is insufficiently specified. – André Nicolas Jun 23 '11 at 2:12
@amWhy: If exactly $40$ are in each of the "exactly two" groups, why would one ask how many passed two? To test the operation $40+40+40$? Conceivably. If a total of $40$ passed exactly two, can't determine how many passed each course. The original problem may have been OK, we are getting a condensed version. – André Nicolas Jun 23 '11 at 2:22

Inclusion-exclusion: Total number of non-absent students equals number passing algebra plus number passing statistics plus number passing mathematics, minus (number passing algebra and stats plus number passing algebra and math plus number passing stats and math), plus number passing all three. Number passing algebra and stats is the 40 who passed all three, plus the $x$ who passed just algebra and stats. You get an equation for $x$.

That stats class must have been a nightmare!

EDIT: This seems to be quite wrong. As others note, it appears there isn't enough information to solve the problem.

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"Total number of non-absent students equals number passing algebra plus number passing statistics plus number passing mathematics, minus (number passing algebra and stats plus number passing algebra and math plus number passing stats and math), plus number passing all three." That is not the total number of non-absent students, but the number of non-absent students passing at least one class. – user12205 Jun 23 '11 at 0:42
@Jeroen, yes, you're right. – Gerry Myerson Jun 23 '11 at 1:48

Let me answer this question with a question, because Gerry's answer does not look like a correct one to me (or I am just confused by the original post).

Denote the total number of non-absent students that pass at least one course with a, the number of students passing Algebra as $|A|$, the number of students passing Stat as $|S|$ and the number of students passing Mathematics as $|M|$.

We know $|A \cap M| = |A \cap S| = |S \cap M| = x$.

Also

$a = |A| + |M| + |S| - 3x + |A \cap S \cap M| = 490 - 3x$

Now we are stuck because we have no information about students failing? Or did I misinterpret the original problem?

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Valen: I tried to think of possible interpretations of the somewhat imprecise wording that would make the problem solvable. I don't think there are any. That's why a while ago I wrote a comment under the question, asking for the number of people who failed everything. – André Nicolas Jun 23 '11 at 1:36
@user6312 I don't see any comments asking for the number of people who failed everything. Did you delete it? Edit: now I see what you mean! I thought your question wasn't serious because of the "not a great class"-begin. My bad :) – user12205 Jun 23 '11 at 1:44
I think $x$ is $|A\cap M\cap S^c|$, the number passing algebra and math and not statistics. – Gerry Myerson Jun 23 '11 at 1:51
@Gerry, thanks, I did accidentally write union instead of intersection. I wanted $x$ to be the number of students that pass on two courses. – user12205 Jun 23 '11 at 2:07