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In a group of 68 students, each student is registered for at

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

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New post 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?

A. 13
B. 10
C. 9
D. 8
E. 7
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Re: MGMAT CAT 3 Overlapping Sets  [#permalink]

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New post 18 Dec 2010, 12:20
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tonebeeze wrote:
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}:

68 = 25 + 25 + 34 - {people in exactly 2 groups} - 2*3 + 0 --> {people in exactly 2 groups}=10

Answer: B.

Look at the diagram:
Attachment:
untitled.PNG
untitled.PNG [ 5.66 KiB | Viewed 46823 times ]
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}.

For more on this check: formulae-for-3-overlapping-sets-69014.html#p729340

As for your question: yes, questions on 3 overlapping sets are quite common for the GMAT.

Hope it helps.
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Re: MGMAT CAT 3 Overlapping Sets  [#permalink]

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New post 18 Dec 2010, 12:27
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Hi!

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.
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Re: MGMAT CAT 3 Overlapping Sets  [#permalink]

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New post 18 Dec 2010, 13:15
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.
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Re: MGMAT CAT 3 Overlapping Sets  [#permalink]

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New post 07 Sep 2012, 07:42
Can this question be done with the help if matrix?
If yes, then please let me know the procedure.
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Re: MGMAT CAT 3 Overlapping Sets  [#permalink]

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New post 07 Sep 2012, 12:05
siddharthasingh wrote:
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.
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Re: In a group of 68 students, each student is registered for at  [#permalink]

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New post 07 Sep 2012, 12:27
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Re: In a group of 68 students, each student is registered for at  [#permalink]

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

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New post 07 Sep 2012, 13:48
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Re: In a group of 68 students, each student is registered for at  [#permalink]

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New post 11 Jun 2013, 08:24
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Overlapping Sets:
advanced-overlapping-sets-problems-144260.html
how-to-draw-a-venn-diagram-for-problems-98036.html

All DS Overlapping Sets Problems to practice: search.php?search_id=tag&tag_id=45
All PS Overlapping Sets Problems to practice: search.php?search_id=tag&tag_id=65

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

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New post 30 Jan 2016, 17:57
Suppose A=all 3
B=History and english
C=history and math
D=math and english
E=english only
F=history only
G=math only

we know that:
A+B+C+D+E+F+G=68
A+B+C+F=25
A+C+D+G=25
A+B+D+E=34
A=3

OK, so we have
B+C+F=22
C+D+G=22
B+D+E=34

add all these 3:
2B+2C+2D+E+F+G=75
we then have:
B+C+D+E+F+G=65

substract from first one the second one:
B+C+D=10
B,C,D - 2 objects only.
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Re: In a group of 68 students, each student is registered for at  [#permalink]

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New post 12 Sep 2016, 13:02
The master information collation on all things Overlapping Sets. Thanks to this(and the authors :P) for making life simpler
overlapping-sets-made-easy-205636.html
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Re: In a group of 68 students, each student is registered for at  [#permalink]

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New post 30 Nov 2016, 02:04
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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
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Re: In a group of 68 students, each student is registered for at  [#permalink]

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New post 09 Feb 2018, 11:53
Hi All,

3-Group Overlapping Sets questions are relatively rare on the Official GMAT (you likely will NOT see this version of Overlapping Sets on Test Day). However, there is a formula that you can use to solve it.

Total = (1st group) + (2nd group) + (3rd group) - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd) - 2(all 3 groups).

In overlapping sets questions, any person who appears in more than one group has been counted more than once. When dealing with groups of people, you're not supposed to count any individual more than once, so the formula 'subtracts' all of the extra times that a person is counted. For example, someone who is in BOTH the 1st group and the 2nd group will be counted twice....that's why we SUBTRACT that person later on [in the (1st and 2nd) group].

In this prompt, we're given the Total, a number for each of the 3 individual groups and the number of people who appear in all 3 groups. The equation would look like this...

68 = 25 + 25 + 34 - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd)- 2(3)

68 = 84 - 6 - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd)

68 = 78 - (1st and 2nd) - (1st and 3rd) - (2nd and 3rd)

(1st and 2nd) + (1st and 3rd) + (2nd and 3rd) = 10

Since the prompt asks for the total number of students that are in exactly 2 classes, we have our answer.

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

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New post 08 Nov 2018, 10:41
Hi guys, I actually got the right answer for this problem, but wanted to confirm if my method of solving it is actually accurate or if I just "got lucky."

History 25
Math 25
English 34

Subtract 3 from each number is 3 people are in all three classes

History 25-3 = 22
Math 25-3 = 22
English 34-3 = 31

Additionally, subtract 3 from the original 68 number since you want to "get rid of them" while you count for the pool.

68-3 = 65

Add up 22+22+31 to get 75

75-65 = 10

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

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Re: In a group of 68 students, each student is registered for at   [#permalink] 18 Nov 2019, 04:54
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