josh80 wrote:Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?
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Here is a helpful 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 = 59.
Poetry = 22.
History = 27.
Writing = 28.
Exactly 2 of the groups = 6.
Let x = the number in all 3 groups.
Plugging these values into the formula, we get:
59 = 22 + 27 + 28 - 6 - 2x
59 = 71 - 2x
x=6.
The correct answer is
C.
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