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Here is the problem.

70 students are enrolled in Math, English, or German. 40 students are in Math, 35 are in English, 30 are in German. 15 students are enrolled in all 3 of the courses. How many of the students are enrolled in exactly two of the courses: Math, English, and German?

The book has very bad explanation and possibly wrong answer for this problem. Using my own logic I get a different answer than the book.

Please let me know what you guys get as answer for this problem.

70 students are enrolled in Math, English, or German. 40 students are in Math, 35 are in English, 30 are in German. 15 students are enrolled in all 3 of the courses. How many of the students are enrolled in exactly two of the courses: Math, English, and German?

The book has very bad explanation and possibly wrong answer for this problem. Using my own logic I get a different answer than the book.

Please let me know what you guys get as answer for this problem.

thank you.

Try using a Venn diagram... _________________

Best,

AkamaiBrah Former Senior Instructor, Manhattan GMAT and VeritasPrep Vice President, Midtown NYC Investment Bank, Structured Finance IT MFE, Haas School of Business, UC Berkeley, Class of 2005 MBA, Anderson School of Management, UCLA, Class of 1993

AuBuC = A + B + C - ( AnB + BnC + CnA) - 2( AnBnC)

70 = 40 + 35 + 30 - ( x) - 2 ( 15 )

x = 5.

i'm not able to understand why you subtracted "2( AnBnC)". please explain.

i did it by venn diagram.

The formula that tkarthi4u listed is a standard formula for venn diagrams with 3 circles.

If you add A + B + C, you have overlaps that you need to remove to prevent double-counting. AnB, BnC, and CnA overlap and are double-counted, so you need to subtract each of them once. If you subtracted AnB twice, you would not count that section altogether.

(AnBnC) is subtracted twice because when you add A, B, and C, you are triple-counting that area. You need to subtract out that area twice (and thereby only counting that area once).

If you add A + B + C, you have overlaps that you need to remove to prevent double-counting. AnB, BnC, and CnA overlap and are double-counted, so you need to subtract each of them once. If you subtracted AnB twice, you would not count that section altogether.

(AnBnC) is subtracted twice because when you add A, B, and C, you are triple-counting that area. You need to subtract out that area twice (and thereby only counting that area once).

I think the standard formula is AuBuC = A + B + C - ( AnB + BnC + CnA) + ( AnBnC)

(AnBnC) is subtracted twice while adding the third circle, that is why we need to add it Once.

1) (40+35+30) - 70 = 35 - the number of students with two courses and the double number of students with three courses. 2) 35 - 2*15 = 5 - the number of students with two courses. _________________