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In a group of 68 students, each student is registered for at [#permalink]

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18 Dec 2010, 11:55

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In a group of 68 students, each student is registered for at least one of three classes – History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?

What is the quickest method to solve 3-Set, Overlapping Set problems? Are they common on the GMAT (if you are scoring 46+ on Quant)?

In a group of 68 students, each student is registered for at least one of three classes – History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?

a. 13 b. 10 c. 9 d. 8 e. 7

"Each student is registered for at least one of three classes" means that there are no students who are registered for none of the classes.

Total = {people in group A} + {people in group B} + {people in group C} - {people in exactly 2 groups} - 2*{people in exactly 3 groups} + {people in none of the groups}:

We need to find {people in exactly 2 groups}, so yellow section. Now, when we sum {people in group A} + {people in group B} + {people in group C} we count students who are in exactly 2 groups (yellow section) twice, so to get rid of double counting we are subtracting {people in exactly 2 groups} once.

Similarly when we sum {people in group A} + {people in group B} + {people in group C} we count students who are in exactly 3 groups (blue section) thrice (as it is the portion of all three groups), so to count this group only once we are subtracting 2*{people in exactly 3 groups}.

There are two approaches you can take for 3 overlapping set questions. Which one works best for you is really a matter of preference.

Many people find Venn diagrams to be the best approach - draw 3 circles that have both double and triple overlapping sections. Venn diagrams are an excellent tool, especially for visual learners.

On the other hand, if you're an equation kind of guy (or gal), there are two different equations that come in handy:

True # of objects = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in exactly 2 groups) - 2(# in all 3 groups);

and:

True # of objects = (total in exactly 1 group) + (total in exactly 2 groups) + (total in exactly 3 groups).

There are a few other variations of those equations as well. In theory, you should add "+ total in none of the groups" to each equation, but I don't think I've ever seen a 3-set question on the GMAT in which everyone wasn't a member of at least one group.

Even at high levels of the exam, 3 set questions aren't particularly common - most test takers see 0 or 1 of them, rarely 2.

tonebeeze Bunuel's explanation at the link he provides was very helpful for me. I recommend sticking to the equations as the venn diagrams tend to throw me off.

Can this question be done with the help if matrix? If yes, then please let me know the procedure.

The matrix method of 2 X 2 works well for two overlapping sets. For three overlapping sets we would need a 3D matrix of 2 X 2 X 2, which is hard (if not impossible) to visualize on a 2D paper or computer screen.

So, stick with the Venn diagrams or the formulas.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

Re: In a group of 68 students, each student is registered for at [#permalink]

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07 Sep 2012, 13:16

siddharthasingh wrote:

Why can't a 3x3 matrix be used here?

We have 3 types of classes, call them A, B, C. Each student can be or not in A - 2 possibilities, can be or not in B, also 2 possibilities, can be or not in C, another two possibilities. So there is a total of 2 x 2 x 2 = 8 different subsets and not 3 x 3 = 9.

If for two overlapping sets A and B you would use a 2 x 2 matrix (A nonA, B nonB), you don't have where to put the information regarding the third set C. Imagine a cube of 2 X 2 X 2, such that you add the third characteristic related to C on the vertical axis, above the base of 2 x 2 for A and B. The 3D cube of dimensions 2 X 2 x 2 for three overlapping sets is the parallel of the 2D matrix of dimensions 2 X 2 for two overlapping sets.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

Similarly when we sum {people in group A} + {people in group B} + {people in group C} we count students who are in exactly 3 groups (blue section) thrice (as it is the portion of all three groups), so to count this group only once we are subtracting 2*{people in exactly 2 groups}.

Bunuel, did you mean that we subtract 2*{people in exactly 3 groups} right? not people in exactly 2 groups....

Similarly when we sum {people in group A} + {people in group B} + {people in group C} we count students who are in exactly 3 groups (blue section) thrice (as it is the portion of all three groups), so to count this group only once we are subtracting 2*{people in exactly 2 groups}.

Bunuel, did you mean that we subtract 2*{people in exactly 3 groups} right? not people in exactly 2 groups....

Re: In a group of 68 students, each student is registered for at [#permalink]

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22 Nov 2014, 08:20

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Re: In a group of 68 students, each student is registered for at [#permalink]

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19 Feb 2016, 18:40

Hi Brunel, I am also new to Venn Diagrams - can I calculate this problem as 68-3-(25-3)-(25-3)-(34-3)=-10? Even though the answer is negative, if I transform it to positive - seems to be correct

Re: In a group of 68 students, each student is registered for at [#permalink]

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12 Sep 2016, 13:02

The master information collation on all things Overlapping Sets. Thanks to this(and the authors ) for making life simpler overlapping-sets-made-easy-205636.html _________________

Re: In a group of 68 students, each student is registered for at [#permalink]

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30 Nov 2016, 02:04

tonebeeze wrote:

In a group of 68 students, each student is registered for at least one of three classes – History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?

A. 13 B. 10 C. 9 D. 8 E. 7

Solution: Let student who attended exactly 1 course= X, 2 courses =Y, 3 courses = Z (GIVEN Z=3)

Total = X+Y+Z 68= X+Y+3 X+Y=65 ---------(1)

SUM of all the courses :

h + m + e = X +2Y +3Z : Reason : every course considered common between 2 is repeated 2 times, while Z is repeated 3 times : each times whenever we take 2 courses together ( x is in anb , bnc, anc) 25+25+34 = X+2Y +3(3) 84-9 = X+2Y X+2Y = 75 X+Y=65 FROM EQUATION (1) solving this we get : Y=10