One formula for 3 overlapping groups:
T = Group 1 + Group 2 + Group 3 - (exactly 2 groups) - 2(all 3 groups)
The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in the three groups:
Those in exactly 2 of groups are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups 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.
VJesus12 wrote:There are 68 children in the cafeteria of a school and all of the children have something for lunch. Thirty-four of the children brought lunches from home, 23 of the children bought a drink from the cafeteria beverage machine, and 32 of the children bought fruit in the cafeteria. If 18 children did at least 2 of these things, how many children did exactly two of these things?
(A) 3
(B) 6
(C) 9
(D) 13
(E) 15
In the problem above:
Group 1 = Lunch = 34.
Group 2 = Drink = 23.
Group 3 = Fruit = 32.
Plugging these values in the formula above, we get:
T = 34 + 23 + 32 - (exactly 2) - 2(all 3)
T = 89 - (exactly 2) - 2(all 3)
We can PLUG IN THE ANSWERS, which represent the number of children who did exactly 2 of the 3 things.
When the correct answer is plugged in, T = 68.
B: Exactly 2 things = 6, implying that all 3 things = 12, for a total of 18 students who did at least 2 of the 3 things
In this case:
T = 89 - 6 - 2(12) = 59.
The value of T is too small.
Eliminate B.
D: Exactly 2 things = 13, implying that all 3 things = 15, for a total of 18 students who did at least 2 of the 3 things
In this case:
T = 89 - 13 - 2(5) = 66.
The value of T is still too small.
Eliminate D.
The results above indicate the following:
The greater the answer choice, the greater the value of T.
Thus:
For the value of T to increase from 66 to 68, a greater answer choice is needed
The correct answer is
E.
E: Exactly 2 things = 15, implying that all 3 things = 3, for a total of 18 students who did at least 2 of the 3 things
In this case:
T = 89 - 15 - 2(3) = 68.
Success!
Check here for another problem about triple-overlapping groups:
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