In a group of 68 students, each student is registered for at least one of three classes - History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?
(A)13
(B)10
(C) 9
(D) 8
(E) 7
This problem is testing your knowledge of overlapping groups. Here is a formula for 3 overlapping groups in which sometimes 2 groups overlap and sometimes all 3 groups overlap:
T = G1 + G2 + G3 - (those in 2 of the groups) - 2*(those in all 3 groups)
The big idea with overlapping groups is to
subtract the overlaps. When we count everyone in the 3 groups, those in 2 of the groups will be counted twice, so they need to subtracted from the total
once. Those in all 3 groups will be counted 3 times, so they need to be subtracted from the total
twice.
In the problem above:
T = 68
G1+G2+G3 = history + math + english = 25+25+34 = 84
Those registered for exactly 2 subjects = x
Those registered for all 3 subjects = 3
Plugging into the formula, we get:
68 = 84 - x - 2*3
68 = 78 - x
x = 10.
The correct answer is
B.
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