The fastest method to solve this question

[pavitkumar]

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?

. 13
.10
.9
8
7

OA 10

Source MGAMT

I got the explanation given by manhattan but its way too long and i find it difficult finishing the question using that method in 2/2.5 minutes. Thus I want to know what is the best approach for this question.

[cindy79]

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?

. 13
.10
.9
8
7

remember following formula:
︱A∪B∪C︱�︱A︱＋︱B︱＋︱C︱�︱A∩B︱�︱B∩C︱�︱C∩A︱＋︱A∩B∩C︱
then u can calculate the answer.

[aneesh.kg]

Well, just a small modification to what Cindy wrote. What she wrote is correct, but the below (alternate) formula will give you the answer sooner:

︱A∪B∪C︱�︱A︱＋︱B︱＋︱C︱� [The number of people in exactly two groups] - 2 *︱A∩B∩C︱

68 = 25 + 25 + 34 - x - 2*3

x = 10

Note: It's useful to know both the formulae. And, it's even more useful to know where they come from.

[pavitkumar]

︱A∪B∪C︱�︱A︱＋︱B︱＋︱C︱� [The number of people in exactly two groups] - 2 *︱A∩B∩C︱

68 = 25 + 25 + 34 - x - 2*3

Hey Aneesh, as you said its important to know, "where they come from"

I didn't get why your multiplied 3 ( i.e. A∩B∩C ) with 2 before subtracting.

Thank you guys for your time.

[aneesh.kg]

Um, Draw a Venn Diagram (the three overlapping circles) to understand this.
If you name the three groups as A, B and C

|A U B U C| is all the area inside those circles.

so,

|A U B U C| = |A| + |B| + |C| - (what else?)

Since, the question is talking about the region for exactly two groups, let's subtract that region from the equation.

so,

|A U B U C| = |A| + |B| + |C| - [People in exactly two groups] -/+ (anything else?)

well now,

when we added the three groups A + B + C, we have counted the ︱A∩B∩C︱ region thrice. But we want it only once. So, WE WILL SUBTRACT IT TWICE.

So, now the final equation is

|A U B U C| = |A| + |B| + |C| - [People in exactly two groups] - 2*︱A∩B∩C︱

makes sense?

[GMATGuruNY]

I posted an explanation here:

https://www.beatthegmat.com/set-prob-again-t76727.html

[ronnie1985]

Use venn diagram to solve

[amit28it]

You can solve it in this manner and in my opinion its the best and fastest way to solve it,
A∪B∪C︱�︱A︱＋︱B︱＋︱C︱� [people in exactly two groups] - 2 *︱A∩B∩C︱

68 = 25 + 25 + 34 - x - 2*3

x = 10
Factoring Trinomials