alex.gellatly wrote:Sets A, B, and C have some elements in common. If 16 elements are in both A and B, 17 elements are in both A and C, and 18 elements are in both B and C, how many elements do all three of the sets A, B, and C have in common?
(1) Of the 16 elements that are in both A and B, 9 elements are also in C.
(2) A has 25 elements, B has 30 elements, and C has 35 elements.
Thanks
Here is the formula for triple-overlapping groups A, B and C:
T = A + B + C - (AB + AC + BC) - 2(ABC)
The big idea with overlapping group problems is to
subtract the overlap.
When we count all of the elements in TWO of the groups (AB + BC + AC), these elements get counted TWICE.
So that these elements are not double-counted, they are subtracted from the total ONCE.
When we count all of the elements in all THREE groups (ABC), these elements get counted THREE TIMES.
So that these elements are not triple-counted, they are subtracted from the total TWICE.
It is given in the question stem that AB + AC + BC = 16+17+18 = 51.
Question: What is ABC?
Statement 1: Of the 16 elements that are in both A and B, 9 elements are also in C.
The 9 elements in both A and B that are also in C are in ALL THREE GROUPS.
Thus, ABC = 9.
SUFFICIENT.
Statement 2: A=25, B=30, C=35
Plugging these values and AB+AC+BC=51 into the formula above, we get:
T = 25+30+35 - 51 - 2(ABC)
T = 39 - 2(ABC)
Without the total number of elements, we can't determine the value of ABC.
INSUFFICIENT.
The correct answer is
A.
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