aditiniyer wrote:8) At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry, and English. 53 students study math, 88 study chemistry, and 58 study English. If 6 students take all 3 classes, how many take exactly 2 classes?
A) 37
B) 43
C) 45
D) 60
E) 70
Here is a useful 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:
T = 150.
Math = 53.
Chemistry = 88.
English = 58.
Let exactly 2 of the groups = x.
All 3 groups = 6.
Plugging these values into the formula, we get:
150 = 53 + 88 + 58 - x - 2*6
150 = 187 - x
x = 37.
The correct answer is
A.
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