Here is one 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 = 90.
A = high jump = 20.
B = long jump = 40.
C = 100-meter dash = 60.
Since the number of students participating in exactly 2 tryouts is unknown, AB + AC + BC = x.
Since 5 students participate in all 3 tryouts, ABC = 5.
Plugging these values into the formula, we get:
90 = 20 + 40 + 60 - x - 2(5)
90 = 110 - x
x = 20.
The correct answer is
B.
Check here for another problem about triple-overlapping groups:
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