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.
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