Of a group of people, 10 play piano, 11 play guitar

guerrero wrote:Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3
B. 6
C. 9
D. 12
E. 15

OAB



How to solve it using the formula for 3 overlapping groups:

T = A + B + C - (AB + AC + BC) - 2(ABC)

Not sure how to solve the same question using the formula, but by Venn Diagram its easy. However, you need to understand what exactly you have to calculate. This question, at times, can give you wrong conception that you only need to find value for R1+ R2 + R3 (refer attachment); infact, you need to find out R1 + R2 + R3 + R4 (you already know value for R4 = 3).
So.
(A + R1 + R4 + R2) + (B + R1 + R3 + R4) + (C + R2 + R4 + R3) = 10 + 11 + 14
20 + 3(R4) + 2(R1 + R2 + R3) = 35
R1 + R2 + R3 = 3
Hence, R1 + R2 + R3 + R4 = 6

IMP: The question DOESN’T say that you need to find people WHO ONLY PLAY TWO.

Brent@GMATPrepNow wrote:

guerrero wrote:Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3
B. 6
C. 9
D. 12
E. 15

I'd use a diagram to solve this question.
First place all of the given information on the diagram.


Our goal is to find the sum: a + b + c [the total number of people who play exactly 2 instruments]

We can write a separate equation for the people who play each instrument.
Violin: a + b + x + 3 = 14
Guitar: a + c + y + 3 = 11
Piano: b + c + z + 3 = 10

Add all three equations to get: 2(a + b + c) + (x + y + z) + 9 = 35
Since we're told that 20 play only one instrument, we know that (x + y + z) = 20
So, we now have: 2(a + b + c) + 20 + 9 = 35
Simplify: 2(a + b + c) = 6
Solve: (a + b + c) = 3

So, 3 people play exactly 2 instruments.
Answer: A

Cheers,
Brent

Here's one possible scenario:

Brent, the question doesn't say that we need to find people who plays EXACTLY 2 instruments. So, I believe we need to consider those 3 people who played all 3 instruments.

Correct me if I am wrong!

guerrero wrote:Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play at least 2 instruments?

A. 3
B. 6
C. 9
D. 12
E. 15

OAB

To make the intent of the question stem clear, I've added the phrase in red.

Here is the formula for 3 overlapping groups:

T = A + B + C - (AB + AC + BC) - 2(ABC)

The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of the groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.

In the problem above:
Let x = the number who play exactly 2 instruments.
T = number who play exactly 1 instrument + number who play exactly 2 instruments + number who play all 3 instruments = 20+x+3.
Piano = 10.
Guitar = 11.
Violin = 14.
Exactly 2 instruments = x.
All 3 instruments = 3.

Plugging these values into the formula, we get:
20 + x + 3 = 10 + 11 + 14 - x - 2(3)
x + 23 = 29 - x
2x = 6
x = 3.

Thus;
Total who play at least 2 instruments = number who play exactly 2 instruments + number who play all 3 instruments = 3+3 = 6.

The correct answer is B.